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Gravitationally lensed curved arcs provide a wealth of information about the underlying lensing distortions. Extracting precise lensing information from extended sources is a key component in many studies aiming to answer fundamental questions about the Universe. To maintain accuracy with increased precision, it is of vital importance to characterize and understand the impact of degeneracies inherent in lensing observables. In this work, we present a formalism to describe the gravitational lensing distortion effects resulting in curved extended arcs based on the eigenvectors and eigenvalues of the local lensing Jacobian and their directional differentials. We identify a non-local and non-linear extended deflector basis that inherits these local properties. Our parameterization is tightly linked to observable features in extended sources and allows one to accurately extract the lensing information of extended images without imposing an explicit global deflector model. We quantify what degeneracies can be broken based on specific assumptions on the local lensing nature and assumed intrinsic source shape. Our formalism is applicable from the weak linear regime, the semi-linear regime all the way up to the highly non-linear regime of highly magnified arcs of multiple images. The methodology and implementation presented in this work provides a framework to assessing systematics, to guide inference efforts in the right choices in complexity based on the data at hand, and to quantify the lensing information extracted in a model-independent way., Comment: 25 pages, 11 Figures, code repository: https://github.com/sibirrer/curved_arcs accepted for publishing in AAS journal

We give a physical interpretation of the formalism intrinsic degeneracies of the gravitational lensing formalism that we derived on a mathematical basis in part IV of this series. We find that all degeneracies occur due to the partition of the mass density along the line of sight. Usually, it is partitioned into a background (cosmic) density and a foreground deflecting object. The latter can be further partitioned into a main deflecting object and perturbers. Weak deflecting objects along the line of sight are also added, either to the deflecting object or as a correction of the angular diameter distances, perturbing the cosmological background density. A priori, this is an arbitrary choice of reference frame and partition. They can be redefined without changing the lensing observables which are sensitive to the integrated deflecting mass density along the entire line of sight. Reformulating the time delay equation such that this interpretation of the degeneracies becomes easily visible, we note that the source can be eliminated from this formulation, which simplifies reconstructions of the deflecting mass distribution or the inference of the Hubble constant, $H_0$. Subsequently, we list necessary conditions to break the formalism intrinsic degeneracies and discuss ways to break them by model choices or including non-lensing observables like velocity dispersions along the line of sight with their advantages and disadvantages. We conclude with a systematic summary of all formalism intrinsic degeneracies and possibilities to break them., 13 pages, 3 figures, accepted for publication in MNRAS, comments very welcome

We investigate a physical, composite alignment model for both spiral and elliptical galaxies and its impact on cosmological parameter estimation from weak lensing for a tomographic survey. Ellipticity correlation functions and angular ellipticity spectra for spiral and elliptical galaxies are derived on the basis of tidal interactions with the cosmic large-scale structure and compared to the tomographic weak lensing signal. We find that elliptical galaxies cause a contribution to the weak-lensing dominated ellipticity correlation on intermediate angular scales between $\ell\simeq40$ and $\ell\simeq400$ before that of spiral galaxies dominates on higher multipoles. The predominant term on intermediate scales is the negative cross-correlation between intrinsic alignments and weak gravitational lensing (GI-alignment). We simulate parameter inference from weak gravitational lensing with intrinsic alignments unaccounted; the bias induced by ignoring intrinsic alignments in a survey like Euclid is shown to be several times larger than the statistical error and can lead to faulty conclusions when comparing to other observations. The biases generally point into different directions in parameter space, such that in some cases one can observe a partial cancellation effect. Furthermore, it is shown that the biases increase with the number of tomographic bins used for the parameter estimation process. We quantify this parameter estimation bias in units of the statistical error and compute the loss of Bayesian evidence for a model due to the presence of systematic errors as well as the Kullback-Leibler divergence to quantify the distance between the true model and the wrongly inferred one.

We determine the cosmic expansion rate from supernovae of type Ia to set up a data-based distance measure that does not make assumptions about the constituents of the universe, i.e. about a specific parametrisation of a Friedmann cosmological model. The scale, determined by the Hubble constant $H_0$, is the only free cosmological parameter left in the gravitational lensing formalism. We investigate to which accuracy and precision the lensing distance ratio $D$ is determined from the Pantheon sample. Inserting $D$ and its uncertainty into the lensing equations for given $H_0$, esp. the time-delay equation between a pair of multiple images, allows to determine lens properties, esp. differences in the lensing potential ($\Delta \phi$), without specifying a cosmological model. We expand the luminosity distances into an analytic orthonormal basis, determine the maximum-likelihood weights for the basis functions by a globally optimal $\chi^2$-parameter estimation, and derive confidence bounds by Monte-Carlo simulations. For typical strong lensing configurations between $z=0.5$ and $z=1.0$, $\Delta \phi$ can be determined with a relative imprecision of 1.7%, assuming imprecisions of the time delay and the redshift of the lens on the order of 1%. With only a small, tolerable loss in precision, the model-independent lens characterisation developed in this paper series can be generalised by dropping the specific Friedmann model to determine $D$ in favour of a data-based distance ratio. Moreover, for any astrophysical application, the approach presented here, provides distance measures for $z\le2.3$ that are valid in any homogeneous, isotropic universe with general relativity as theory of gravity., Comment: 16 pages, 4 figures, accepted for publication in MNRAS, comments welcome

Based on the standard gravitational lensing formalism with its effective, projected lensing potential in a given background cosmology, we investigate under which transformations of the source position and of the deflection angle the observable properties of the multiple images, i.e. the time delay differences, the relative image positions, relative shapes, and magnification ratios, remain invariant. As these observables only constrain local lens properties, we derive general, local invariance transformations in the areas covered by the multiple images. We show that the known global invariance transformations, e.g. the mass sheet transformation or the source position transformation, are contained in our invariance transformations, when they are restricted to the areas covered by the multiple images and when lens-model-based degeneracies are ignored, like the freedom to add or subtract masses in unconstrained regions without multiple images. Hence, we have identified the general class of invariance transformations that can occur, in particular in our model-independent local characterisation of strong gravitational lenses., 10 pages, 4 figures, in press in A&A, comments very welcome (update to accepted and improved version)

In this paper we apply the symplectic formalism to describe light deflection in general relativity as a scattering process. In the first part of the paper we present gravitational lensing in the action principle and in the Hamiltonian formulations. After defining the Souriau’s symplectic scattering and introducing the mathematical properties in the infinitesimal case, we show that this formalism is an appropriate tool to study the deflection of a light signal in a gravitational field, not only in determining the scattering angle, but also the corresponding redshift and the Shapiro time delay. As examples we apply the symplectic scattering to lensing in weak fields and in the Swiss Cheese cosmological model. The conditions for the applicability of the infinitesimal scattering are derived. Finally we discuss its relation with the non infinitesimal case, we propose an extension of the results of the infinitesimal scattering and examine some properties of the non infinitesimal scattering in the hamiltonian formalism paying attention to the definition of photon sphere.

We measure the cross-correlation between Fermi-LAT gamma-ray photons and over 1000 deg$^2$ of weak lensing data from the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS), the Red Cluster Sequence Lensing Survey (RCSLenS), and the Kilo Degree Survey (KiDS). We present the first measurement of tomographic weak lensing cross-correlations and the first application of spectral binning to cross-correlations between gamma rays and weak lensing. The measurements are performed using an angular power spectrum estimator while the covariance is estimated using an analytical prescription. We verify the accuracy of our covariance estimate by comparing it to two internal covariance estimators. Based on the non-detection of a cross-correlation signal, we derive constraints on weakly interacting massive particle (WIMP) dark matter. We compute exclusion limits on the dark matter annihilation cross-section $\langle\sigma_\rm{ann} v \rangle$, decay rate $\Gamma_\rm{dec}$, and particle mass $m_\rm{DM}$. We find that in the absence of a cross-correlation signal, tomography does not significantly improve the constraining power of the analysis. Assuming a strong contribution to the gamma-ray flux due to small-scale clustering of dark matter and accounting for known astrophysical sources of gamma rays, we exclude the thermal relic cross-section for particle masses of $m_\rm{DM}\lesssim 20$ GeV., Comment: 19 pages, 17 figures, accepted in MNRAS

In this paper, we study gravitational lensing of magnetically charged black hole of string theory as a strong field approximation for the supermassive black hole at the center of NGC4486B. We evaluate light deflection angle numerically, from which we obtain magnifications, Einstein rings and observables for the relativistic images. Finally, we explore time delay between relativistic images when they are on the same as well as opposite side of the lens. It is concluded that charge parameter plays a prominent role in the strong gravitational lensing.

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 may be even more affected by intrinsic correlations than galaxy sizes, making this a bigger systematic for measurements of the weak lensing signal. We also quantify how these intrinsic correlations are affected by sample selection criteria based on sizes and magnitudes., Comment: 17 pages, 12 figures, 1 table; published on MNRAS

We demonstrate a new method to constrain gravity on the largest cosmological scales by combining measurements of cosmic microwave background (CMB) lensing and the galaxy velocity field. $E_G$ is a statistic, constructed from a gravitational lensing tracer and a measure of velocities such as redshift-space distortions (RSD), that can discriminate between gravity models while being independent of clustering bias and $\sigma_8$. While traditionally, the lensing field for $E_G$ has been probed through galaxy lensing, CMB lensing has been proposed as a more robust tracer of the lensing field for $E_G$ at higher redshifts while avoiding intrinsic alignments. We perform the largest-scale measurement of $E_G$ ever, up to 150 Mpc/$h$, by cross-correlating the Planck CMB lensing map with the Sloan Digital Sky Survey III (SDSS-III) CMASS galaxy sample and combining this with our measurement of the CMASS auto-power spectrum and the RSD parameter $\beta$. We report $E_G(z=0.57)=0.243\pm0.060$ (stat) $\pm0.013$ (sys), a measurement in tension with the general relativity prediction at a level of 2.6$\sigma$. Note that our $E_G$ measurement deviates from GR only at scales greater than 80 Mpc/$h$, scales which have not been probed by previous $E_G$ tests. Upcoming surveys, which will provide an order-of-magnitude reduction in statistical errors, can significantly constrain alternative gravity models when combined with better control of systematics., Comment: 12 pages, 11 figures, matches published version in MNRAS, minor edits

This is the fourth in a series of papers studying the astrophysics and cosmology of massive, dynamically relaxed galaxy clusters. Here, we use measurements of weak gravitational lensing from the Weighing the Giants project to calibrate Chandra X-ray measurements of total mass that rely on the assumption of hydrostatic equilibrium. This comparison of X-ray and lensing masses provides a measurement of the combined bias of X-ray hydrostatic masses due to both astrophysical and instrumental sources. Assuming a fixed cosmology, and within a characteristic radius (r_2500) determined from the X-ray data, we measure a lensing to X-ray mass ratio of 0.96 +/- 9% (stat) +/- 9% (sys). We find no significant trends of this ratio with mass, redshift or the morphological indicators used to select the sample. In accordance with predictions from hydro simulations for the most massive, relaxed clusters, our results disfavor strong, tens-of-percent departures from hydrostatic equilibrium at these radii. In addition, we find a mean concentration of the sample measured from lensing data of c_200 = $3.0_{-1.8}^{+4.4}$. Anticipated short-term improvements in lensing systematics, and a modest expansion of the relaxed lensing sample, can easily increase the measurement precision by 30--50%, leading to similar improvements in cosmological constraints that employ X-ray hydrostatic mass estimates, such as on Omega_m from the cluster gas mass fraction., 13 pages. Submitted to MNRAS. Comments welcome

If really such objects like cosmological black holes exist they may be studied with a standard technique like strong and weak gravitational lensing. Cosmological voids can be explained as the result the collapse of large perturbations into black hole with masses of the order of 1014 M ⊙ and the expansion of the universe. The resulting image of the universe is that it is more homogeneous than expected from present observations. In this paper we discuss some lensing properties related to the cosmological black holes (CBHs), namely we consider differences in gravitational lensing for point like mass and extended mass distributions. We consider the singular isothermal sphere model as a toy (illustrative) model for an extended distribution of dark matter and a slightly more complicated isothermal sphere with a core.

We investigate the strong gravitational lensing for black hole with scalar charge in massive gravity. We find that the scalar charge and the type of the black hole significantly affect the radius of the photon sphere, deflection angle, angular image position, angular image separation, relative magnifications and time delay in strong gravitational lensing. Our results can be reduced to that of the Schwarzschild and Reissner-Nordstr$\ddot{o}$m black holes in some special cases., 14 pages, 4 figures, 3 tables

We study the statistics of peaks in a weak lensing reconstructed mass map of the first 450 square degrees of the Kilo Degree Survey. The map is computed with aperture masses directly applied to the shear field with an NFW-like compensated filter. We compare the peak statistics in the observations with that of simulations for various cosmologies to constrain the cosmological parameter $S_8 = \sigma_8 \sqrt{\Omega_{\rm m}/0.3}$, which probes the ($\Omega_{\rm m}, \sigma_8$) plane perpendicularly to its main degeneracy. We estimate $S_8=0.750\pm0.059$, using peaks in the signal-to-noise range $0 \leq {\rm S/N} \leq 4$, and accounting for various systematics, such as multiplicative shear bias, mean redshift bias, baryon feedback, intrinsic alignment, and shear-position coupling. These constraints are $\sim25\%$ tighter than the constraints from the high significance peaks alone ($3 \leq {\rm S/N} \leq 4$) which typically trace single-massive halos. This demonstrates the gain of information from low-S/N peaks. However we find that including ${\rm S/N} < 0$ peaks does not add further information. Our results are in good agreement with the tomographic shear two-point correlation function measurement in KiDS-450. Combining shear peaks with non-tomographic measurements of the shear two-point correlation functions yields a $\sim20\%$ improvement in the uncertainty on $S_8$ compared to the shear two-point correlation functions alone, highlighting the great potential of peaks as a cosmological probe., Comment: 20 pages, 17 figures, accepted for publication in MNRAS

Cosmological parameter constraints from observations of time-delay lenses are becoming increasingly precise. However, there may be significant bias and scatter in these measurements due to, among other things, the so-called mass-sheet degeneracy. To estimate these uncertainties, we analyze strong lenses from the largest EAGLE hydrodynamical simulation. We apply a mass-sheet transformation to the radial density profiles of lenses, and by selecting lenses near isothermality, we find that the bias on H0 can be reduced to 5% with an intrinsic scatter of 10%, confirming previous results performed on a different simulation data set. We further investigate whether combining lensing observables with kinematic constraints helps to minimize this bias. We do not detect any significant dependence of the bias on lens model parameters or observational properties of the galaxy, but depending on the source--lens configuration, a bias may still exist. Cross lenses provide an accurate estimate of the Hubble constant, while fold (double) lenses tend to be biased low (high). With kinematic constraints, double lenses show bias and intrinsic scatter of 6% and 10%, respectively, while quad lenses show bias and intrinsic scatter of 0.5% and 10%, respectively. For lenses with a reduced $\chi^2 > 1$, a power-law dependence of the $\chi^2$ on the lens environment (number of nearby galaxies) is seen. Lastly, we model, in greater detail, the cases of two double lenses that are significantly biased. We are able to remove the bias, suggesting that the remaining biases could also be reduced by carefully taking into account additional sources of systematic uncertainty., Comment: 21 pages, 13 figures, 9 tables; revised version submitted to MNRAS

One of the most powerful techniques to study the dark sector of the Universe is weak gravitational lensing. In practice, to infer the reduced shear, weak lensing measures galaxy shapes, which are the consequence of both the intrinsic ellipticity of the sources and of the integrated gravitational lensing effect along the line of sight. Hence, a very large number of galaxies is required in order to average over their individual properties and to isolate the weak lensing cosmic shear signal. If this `shape noise' can be reduced, significant advances in the power of a weak lensing surveys can be expected. This paper describes a general method for extracting the probability distributions of parameters from catalogues of data using Voronoi cells, which has several applications, and has synergies with Bayesian hierarchical modelling approaches. This allows us to construct a probability distribution for the variance of the intrinsic ellipticity as a function of galaxy property using only photometric data, allowing a reduction of shape noise. As a proof of concept the method is applied to the CFHTLenS survey data. We use this approach to investigate trends of galaxy properties in the data and apply this to the case of weak lensing power spectra., Comment: Accepted for publication in MNRAS

The smaller the angular scales on which the anisotropies of the cosmic microwave background (CMB) are probed the more important their distortion due to gravitational lensing becomes. Here we investigate the maxima and minima of the CMB lensing deflection field using general extreme value statistics. Since general extreme value statistics applies to uncorrelated data in first place we consider appropriately low-pass filtered deflection maps. Besides the suppression of correlations filtering is required for another reason: The lensing field itself is not directly observable but needs to be (statistically) reconstructed from the lensed CMB by means of a quadratic estimator. This reconstruction, though, is noise dominated and therefore requires smoothing, too. In idealized Gaussian realizations as well as in realistically reconstructed data we find that both maxima and minima of the deflection angle components follow consistently a general extreme value distribution of Weibull-type. However, its shape, location and scale parameters vary significantly between different realizations. The statistics' potential power to constrain cosmological models appears therefore rather limited.

We consider effects due to gravitational lensing by hypothetical cosmic strings. We briefly review facts concerning infinite straight strings, obtain the equation for a gravitational lens produced by an infinite string in the classical form, and analyze features of the gravitational lensing for a given object in detail. We consider a finite straight string. We present caustics and critical curves for strings with different lengths and also some images produced by the considered gravitational lens. We propose a method for constructing brightness curves numerically. As an example of the working capacity of this method, we construct a concrete brightness curve for certain gravitational lens parameters.

We have developed a new technique called Direct Shear Mapping (DSM) to measure gravitational lensing shear directly from observations of a single background source. The technique assumes the velocity map of an un-lensed, stably-rotating galaxy will be rotationally symmetric. Lensing distorts the velocity map making it asymmetric. The degree of lensing can be inferred by determining the transformation required to restore axisymmetry. This technique is in contrast to traditional weak lensing methods, which require averaging an ensemble of background galaxy ellipticity measurements, to obtain a single shear measurement. We have tested the ecacy of our tting algorithm with a suite of systematic tests on simulated data. We demonstrate that we are in principle able to measure shears as small as 0.01. In practice, we have tted for the shear in very low redshift (and hence un-lensed) velocity maps, and have obtained null result with an error of 0:01. This high sensitivity results from analysing spatially resolved spectroscopic images (i.e. 3D data cubes), including not just shape information (as in traditional weak lensing measurements) but velocity information as well. Spirals and rotating ellipticals are ideal targets for this new technique. Data from any large IFU or radio telescope is suitable, or indeed any instrument with spatially resolved spectroscopy such as SAMI, ALMA, HETDEX and SKA.

We investigate the feasibility of measuring weak gravitational lensing using 21cm intensity mapping with special emphasis on the performance of the planned Square Kilometre Array (SKA). We find that the current design for SKA-Mid should be able to measure the evolution of the lensing power spectrum at z~2-3 using this technique. This will be a probe of the expansion history of the universe and gravity at a unique range in redshift. The signal-to-noise is found to be highly dependent on evolution of the neutral hydrogen fraction in the universe with a higher HI density resulting in stronger signal. With realistic models for this, SKA Phase 1 should be capable of measuring the lensing power spectrum and its evolution. The signal-to-noise's dependence on the area and diameter of the telescope array is quantified. We further demonstrate the applications of this technique by applying it to two specific coupled dark energy models that would be difficult to observationally distinguish without information from this range of redshift. We also investigate measuring the lensing signal with 21cm emission from the Epoch of Reionization (EoR) using SKA-Low and find that it is unlikely to constrain cosmological parameters because of the small survey size, but could provide a map of the dark matter within a small region of the sky., Comment: 18 pages, 14 figures; version accepted for publication in MNRAS

In this work, we examine the effects of the cosmological constant and a scalar field on the angle of deflection of null geodesics in the equatorial plane of the Dilaton-de Sitter space–time. We employ the Rindler–Ishak method to compute the gravitational lensing. We derive new limits on the value of the cosmological constant, $$\Lambda $$ , based on the bending of light by systems where the lens is a distant galaxy or a cluster of galaxies. These results are essential and important for studies of cosmology.

Gravitational lensing of the CMB is a valuable cosmological signal that correlates to tracers of large-scale structure and acts as a important source of confusion for primordial $B$-mode polarization. State-of-the-art lensing reconstruction analyses use quadratic estimators, which are easily applicable to data. However, these estimators are known to be suboptimal, in particular for polarization, and large improvements are expected to be possible for high signal-to-noise polarization experiments. We develop a method and numerical code, $\rm{LensIt}$, that is able to find efficiently the most probable lensing map, introducing no significant approximations to the lensed CMB likelihood, and applicable to beamed and masked data with inhomogeneous noise. It works by iteratively reconstructing the primordial unlensed CMB using a deflection estimate and its inverse, and removing residual lensing from these maps with quadratic estimator techniques. Roughly linear computational cost is maintained due to fast convergence of iterative searches, combined with the local nature of lensing. The method achieves the maximal improvement in signal to noise expected from analytical considerations on the unmasked parts of the sky. Delensing with this optimal map leads to forecast tensor-to-scalar ratio parameter errors improved by a factor $\simeq 2 $ compared to the quadratic estimator in a CMB stage IV configuration., v3 fixes typos in Eqs. 3.3, 3.4 and footnote 1

Full ray-tracing maps of gravitational lensing, constructed from N-Body simulations, represent a fundamental tool to interpret present and future weak lensing data. However the limitation of computational resources and storage capabilities severely restrict the number of realizations that can be performed in order to accurately sample both the cosmic shear models and covariance matrices. In this paper we present a halo model formalism for weak gravitational lensing that alleviates these issues by producing weak-lensing mocks at a reduced computational cost. Our model takes as input the halo population within a desired light-cone and the linear power spectrum of the underlined cosmological model. We examine the contribution given by the presence of substructures within haloes to the cosmic shear power spectrum and quantify it to the percent level. Our method allows us to reconstruct high-resolution convergence maps, for any desired source redshifts, of light-cones that realistically trace the matter density distribution in the universe, account for masked area and sample selections. We compare our analysis on the same large scale structures constructed using ray-tracing techniques and find very good agreements both in the linear and non-linear regimes up to few percent levels. The accuracy and speed of our method demonstrate the potential of our halo model for weak lensing statistics and the possibility to generate a large sample of convergence maps for different cosmological models as needed for the analysis of large galaxy redshift surveys., 17 pages, 15 figures, accepted for publication in MNRAS

Gravitational lensing information from the two and higher point statistics of the CMB temperature and polarization fields are intrinsically correlated because they are lensed by the same realization of structure between last scattering and observation. Using an analytic model for lens sample covariance, we show that there is one mode, separately measurable in the lensed CMB power spectra and lensing reconstruction, that carries most of this correlation. Once these measurements become lens sample variance dominated, this mode should provide a useful consistency check between the observables that is largely free of sampling and cosmological parameter errors. Violations of consistency could indicate systematic errors in the data and lens reconstruction or new physics at last scattering, any of which could bias cosmological inferences and delensing for gravitational waves. A second mode provides a weaker consistency check for a spatially flat universe. Our analysis isolates the additional information supplied by lensing in a model independent manner but is also useful for understanding and forecasting CMB cosmological parameter errors in the extended $\Lambda$CDM parameter space of dark energy, curvature and massive neutrinos. We introduce and test a simple but accurate forecasting technique for this purpose that neither double counts lensing information nor neglects lensing in the observables., Comment: 14 pages, 9 figures, accepted version

The standard theory of weak gravitational lensing relies on the infinitesimal light beam approximation. In this context, images are distorted by convergence and shear, the respective sources of which unphysically depend on the resolution of the distribution of matter---the so-called Ricci-Weyl problem. In this letter, we propose a strong-lensing-inspired formalism to describe the lensing of finite beams. We address the Ricci-Weyl problem by showing explicitly that convergence is caused by the matter enclosed by the beam, regardless of its distribution. Furthermore, shear turns out to be systematically enhanced by the finiteness of the beam. This implies, in particular, that the Kaiser-Squires relation between shear and convergence is violated, which could have profound consequences on the interpretation of weak lensing surveys., 6 pages, 2 figures, v2: matches published version, some typos corrected

Einstein's theory of General Relativity implies that energy, i.e. matter, curves space-time and thus deforms lightlike geodesics, giving rise to gravitational lensing. This phenomenon is well understood in the case of the Schwarzschild metric, and has been accurately described in the past; however, lensing in the Kerr space-time has received less attention in the literature despite potential practical observational applications. In particular, lensing in such space is not expressible as the gradient of a scalar potential and as such is a source of curl-like signatures and an asymmetric shear pattern. In this paper, we develop a differentiable lensing map in the Kerr metric, reworking and extending previous approaches. By using standard tools of weak gravitational lensing, we isolate and quantify the distortion that is uniquely induced by the presence of angular momentum in the metric. We apply this framework to the distortion induced by a Kerr-like foreground object on a distribution of background of sources. We verify that the new unique lensing signature is orders of magnitude below current observational bounds for a range of lens configurations., Comment: 16 pages, 11 figures

We present the tomographic cross-correlation between galaxy lensing measured in the Kilo Degree Survey (KiDS-450) with overlapping lensing measurements of the cosmic microwave background (CMB), as detected by Planck 2015. We compare our joint probe measurement to the theoretical expectation for a flat $\Lambda$CDM cosmology, assuming the best-fitting cosmological parameters from the KiDS-450 cosmic shear and Planck CMB analyses. We find that our results are consistent within $1\sigma$ with the KiDS-450 cosmology, with an amplitude re-scaling parameter $A_{\rm KiDS} = 0.86 \pm 0.19$. Adopting a Planck cosmology, we find our results are consistent within $2\sigma$, with $A_{\it Planck} = 0.68 \pm 0.15$. We show that the agreement is improved in both cases when the contamination to the signal by intrinsic galaxy alignments is accounted for, increasing $A$ by $\sim 0.1$. This is the first tomographic analysis of the galaxy lensing -- CMB lensing cross-correlation signal, and is based on five photometric redshift bins. We use this measurement as an independent validation of the multiplicative shear calibration and of the calibrated source redshift distribution at high redshifts. We find that constraints on these two quantities are strongly correlated when obtained from this technique, which should therefore not be considered as a stand-alone competitive calibration tool., Comment: 14 pages, 11 figures

In this work, by using strong gravitational lensing (SGL) observations along with Type Ia Supernovae (Union2.1) and gamma ray burst data (GRBs), we propose a new method to study a possible redshift evolution of $\gamma(z)$, the mass density power-law index of strong gravitational lensing systems. In this analysis, we assume the validity of cosmic distance duality relation and the flat universe. In order to explore the $\gamma(z)$ behavior, three different parametrizations are considered, namely: (P1) $\gamma(z_l)=\gamma_0+\gamma_1 z_l$, (P2) $\gamma(z_l)=\gamma_0+\gamma_1 z_l/(1+z_l)$ and (P3) $\gamma(z_l)=\gamma_0+\gamma_1 \ln(1+z_l)$, where $z_l$ corresponds to lens redshift. If $\gamma_0=2$ and $\gamma_1=0$ the singular isothermal sphere model is recovered. Our method is performed on SGL sub-samples defined by different lens redshifts and velocity dispersions. For the former case, the results are in full agreement with each other, while a 1$\sigma$ tension between the sub-samples with low ($\leq 250$ km/s) and high ($>250$ km/s) velocity dispersions was obtained on the ($\gamma_0$-$\gamma_1$) plane. By considering the complete SGL sample, we obtain $\gamma_0 \approx 2$ and $ \gamma_1 \approx 0$ within 1$\sigma$ c.l. for all $\gamma(z)$ parametrizations. However, we find the following best fit values of $\gamma_1$: $-0.085$, $-0.16$ and $-0.12$ for P1, P2 and P3 parametrizations, respectively, suggesting a mild evolution for $\gamma(z)$. By repeating the analysis with Type Ia Supernovae from JLA compilation, GRBs and SGL systems this mild evolution is reinforced., Comment: 11 pages, 5 figures, 1 table, text revised and new analysis included. Accepted for publication in MNRAS

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. Paper I adopts an iterative approach, which degrades toward high redshift. The current paper presents an alternative method, improving over paper I. We prove that the measured galaxy clustering between flux bins allows for simultaneous determination of the lensing power spectrum and the flux dependence of galaxy bias, at this redshift bin. Comparing to paper I, the new approach is not only more straightforward, but also more robust. It identifies an ambiguity in determining the galaxy bias and further discovers a mathematically robust way to suppress this ambiguity to non-negligible level ($\sim 0.1%$). The accurately determined galaxy bias can then be applied to the minimal variance estimator proposed in paper I to improve the lensing map-making. The gain at high redshift is significant. These maps can be used to measure other statistics, such as cluster finding and peak statistics. Furthermore, by including galaxy clustering measurement between different redshift bins, we can also determine the lensing cross power spectrum between these bins, up to a small and correctable multiplicative factor., Comment: 13 pages, 7 figures,submitted to MNRAS

We propose a single galaxy gravitational lensing model with a mass density that has a spiral structure. Namely, we extend the arcsine gravitational lens (a truncated singular isothermal elliptical model), adding an additional parameter that controls the amount of spiraling in the structure of the mass density. An important feature of our model is that, even though the mass density is sophisticated, we succeed in integrating the deflection term in closed form using a Gauss hypergeometric function. When the spiraling parameter is set to zero, this reduces to the arcsine lens.

The microphysical properties of the DM particle can, in principle, be constrained by the properties and abundance of substructures in DM halos, as measured through strong gravitational lensing. Unfortunately, there is a lack of accurate theoretical predictions for the lensing signal of substructures, mainly because of the discreteness noise inherent to N-body simulations. Here we present Recursive-TCM, a method that is able to provide lensing predictions with an arbitrarily low discreteness noise, without any free parameters or smoothing scale. This solution is based on a novel way of interpreting the results of N-body simulations, where particles simply trace the evolution and distortion of Lagrangian phase-space volume elements. We discuss the advantages of this method over the widely used cloud-in-cells and adaptive-kernel smoothing density estimators. Applying the new method to a cluster-sized DM halo simulated in warm and cold DM scenarios, we show how the expected differences in their substructure population translate into differences in the convergence and magnification maps. We anticipate that our method will provide the high-precision theoretical predictions required to interpret and fully exploit strong gravitational lensing observations.

Doppler lensing is the apparent change in object size and magnitude due to peculiar velocities. Objects falling into an overdensity appear larger on its near side, and smaller on its far side, than typical objects at the same redshifts. This effect dominates over the usual gravitational lensing magnification at low redshift. Doppler lensing is a promising new probe of cosmology, and we explore in detail how to utilize the effect with forthcoming surveys. We present cosmological simulations of the Doppler and gravitational lensing effects based on the Millennium simulation. We show that Doppler lensing can be detected around stacked voids or unvirialised over-densities. New power spectra and correlation functions are proposed which are designed to be sensitive to Doppler lensing. We consider the impact of gravitational lensing and intrinsic size correlations on these quantities. We compute the correlation functions and forecast the errors for realistic forthcoming surveys, providing predictions for constraints on cosmological parameters. Finally, we demonstrate how we can make 3-D potential maps of large volumes of the Universe using Doppler lensing., Comment: 17 pages, 14 figures. Minor corrections and improvements; version accepted by MNRAS

The theory of gravitational lensing is well developed for light propagation in vacuum in the approximation of weak deflection. We discuss two ways of extending the studies of usual gravitational lens theory which were recently theoretically developed. One way is to go beyond the weak deflection approximation and consider relativistic images formed by the photons performing one or several revolutions around the central object. Another way is to consider the propagation of light in a plasma instead of vacuum, in the presence of gravity. In this case the lensing angle depends on the plasma distribution and the photon frequency. The gravitational deflection of a light ray even in a homogeneous plasma differs from the vacuum (Einstein) angle and depends on the photon frequency. We also discuss the recent results on the influence of a plasma on positions and magnifications of relativistic images.

The weak lensing effect generates spin-2 distortions, referred to as shear, on the observable shape of distant galaxies, induced by intervening gravitational tidal fields. Traditionally, the spin-2 distortion in the light distribution of distant galaxies is measured in terms of a galaxy ellipticity. This is a very good unbiased estimator of the shear field in the limit that a galaxy is measured at infinite signal-to-noise. However, the ellipticity is always defined as a ratio between two quantities (for example, between the polarisation and measurement of the galaxy size, or between the semi-major and semi-minor axis of the galaxy) and therefore requires some non-linear combination of the image pixels. This means, in any realistic case, this would lead to biases in the measurement of the shear (and hence in the cosmological parameters) whenever noise is present in the image. This type of bias can be understood from the particular shape of the 2D probability distribution of the ellipticity of an object measured from data. Moreover this probability distribution can be used to explore strategies for calibration of noise biases in present and future weak lensing surveys (e.g. KiDS, DES, HSC,Euclid, LSST...)

Strong lensing is an effective way to probing the properties of dark energy. In this paper, we use the strong lensing data to constrain the f(T) theory, which is a new modified gravity to explain the present accelerating cosmic expansion without the need of dark energy. In our discussion, the CMB and BAO data are also added to constrain model parameters tightly and three different f(T) models are studied. We find that strong lensing has an important role on constraining f(T) models, and once the CMB+BAO data is added, a tighter constraint is obtained. However, the consistency of our result with what is obtained from SNIa+CMB+BAO is actually model-dependent.

I show that the lensing masses of the SLACS sample of strong gravitational lenses are consistent with the stellar masses determined from population synthesis models using the Salpeter IMF. This is true in the context of both General Relativity and modified Newtonian dynamics, and is in agreement with the expectation of MOND that there should be little classical discrepancy within the high surface brightness regions probed by strong gravitational lensing. There is also dynamical evidence from this sample supporting the claim that the mass-to-light ratio of the stellar component increases with the velocity dispersion., Comment: 4 pages, 3 figures, table of lenses added, more discussion of lensing with MOND. Accepted for publication in MNRAS, main journal

In the present paper, gravitational lensing is described as the electromagnetic influence from gravity waves on light waves. Previous reports have described the dynamic electromagnetic processes of the atom, the photon and gravity. Results from these reports have been compiled into a theoretical model. The theoretical model describes the mechanism which results in gravitational lensing. The study also displays how the electromagnetic characteristics of gravity waves and light waves and the mechanism creating gravitational lensing are measured.

Subject of this paper is the weak lensing effect on galaxies that show intrinsically correlated ellipticities. In our model, we investigate the distortion of the ellipticity field if the galaxies experience an apparent shift in their position by weak lensing deflection and compare this effect to the shearing effect induced by tidal fields. Starting with a derivation of intrinsic ellipticity spectra by employing a tidal torquing model generating galactic angular momenta, we model the galaxy ellipticity by assuming that the galactic disk forms perpendicularly to the host halo angular momentum direction and derive intrinsic ellipticity E-mode and B-mode spectra from the angular momentum statistics. The lensing effect on the ellipticity field is modeled by employing the methodology developed in the framework of lensing of the cosmic microwave background polarisation. For EUCLID, ellipticity correlations are altered by lensing deflection on multipoles l>1000 by ~5% for the ellipticity E-modes and by ~30% for the B-modes, while a shallower survey would exhibit larger changes on larger angular scales. In addition to the convolving effect of lensing on the ellipticity spectra we investigate the E/B-mode conversion, and discuss the possibility of measuring correlations between different multipoles which is evoked by the homogeneity breaking effect of the lensing displacement. Our conclusion is that although shape correlations generated by weak gravitational shear is dominant, the shifting effect due to lensing is shaping the ellipticity spectra on small angular scales and causes a number of interesting phenomena, which might be observable by future surveys., 10 pages, 7 figures, submitted to MNRAS

We study three-dimensional microlensing where two lenses are located at different distances along the line of sight. We formulate the lens equation in complex notations and recover several previous results. There are in total either 4 or 6 images, with an equal number of images with positive and negative parities. We find that the sum of signed magnifications for six image configurations is unity. Furthermore, we show that the light curves can be qualitatively different from those for binary lensing in a single plane. In particular, the magnifications between a `U'-shaped caustic crossing can be close to unity, rather than having a minimum magnification of 3 as in the single plane binary lensing. There is only a small probability three-dimensional microlensing events will be seen in microlensing toward the Galactic centre. It is more likely they will be first seen in cosmological microlensing., 9 pages, 4 figures, MNRAS, comments welcome!

We investigate the potential of constraining the mass to light ratio of field galaxies using weak lensing shear and flexions. A suite of Monte Carlo simulations are used to generate weak lensing observations with different noise models. Using mock data, we find that the inclusion of flexions can improve the estimate of foreground halo parameters, but the details are strongly dependent on noise in the model. In the intrinsic noise limit, both shear and flexions are promising tools to study the mass to light ratio of galaxies. However, if the noise model of flexions follows the form described by Rowe et al., there is only ~5% improvement in the constraints even with next generation lensing observations .

In this paper we show how intrinsic alignments can be incorporated consistently in the formalism of 3d weak lensing. We use two different descriptions of the intrinsic galaxy ellipticities, the so-called linear and quadratic model, respectively. For both models we derive the covariance matrix of the intrinsic alignment signal (II-alignments) and GI-alignments (the cross correlation of intrinsic and lensing induced ellipticities). Evaluating the covariance matrices for the linear model numerically and comparing the results to the cosmic shear signal we find that for a low redshift survey the total covariance matrix is dominated by the contributions of the II-alignments. For a Euclid-like survey II-alignments still dominate over GI-alignments but they are more than one order of magnitude smaller than the lensing signal. The shape of the ellipticity covariance matrices is quite different in the k-k'-plane for cosmic shear on the one hand and intrinsic alignments on the other hand. In comparison to lensing both alignment types tend to be rather elongated along the diagonal k=k'. Moreover, for high multipoles (l ~ 100) intrinsic alignments are strongly concentrated along that diagonal., Comment: 13 pages, 5 figures, MNRAS accepted

We present a quantitative analysis of the largest contiguous maps of projected mass density obtained from gravitational lensing shear. We use data from the 154 deg2 covered by the Canada-France-Hawaii Telescope Lensing Survey. Our study is the first attempt to quantitatively characterize the scientific value of lensing maps, which could serve in the future as a complementary approach to the study of the dark universe with gravitational lensing. We show that mass maps contain unique cosmological information beyond that of traditional two-points statistical analysis techniques. Using a series of numerical simulations, we first show how, reproducing the CFHTLenS observing conditions, gravitational lensing inversion provides a reliable estimate of the projected matter distribution of large scale structure. We validate our analysis by quantifying the robustness of the maps with various statistical estimators. We then apply the same process to the CFHTLenS data. We find that the 2-points correlation function of the projected mass is consistent with the cosmological analysis performed on the shear correlation function discussed in the CFHTLenS companion papers. The maps also lead to a significant measurement of the third order moment of the projected mass, which is in agreement with analytic predictions, and to a marginal detection of the fourth order moment. Tests for residual systematics are found to be consistent with zero for the statistical estimators we used. A new approach for the comparison of the reconstructed mass map to that predicted from the galaxy distribution reveals the existence of giant voids in the dark matter maps as large as 3 degrees on the sky. Our analysis shows that lensing mass maps can be used for new techniques such as peak statistics and the morphological analysis of the projected dark matter distribution., Version accepted by MNRAS

We study the effects on the number counts of sub-millimetre galaxies due to gravitational lensing. We explore the effects on the magnification cross section due to halo density profiles, ellipticity and cosmological parameter (the power-spectrum normalisation $\sigma_8$). We show that the ellipticity does not strongly affect the magnification cross section in gravitational lensing while the halo radial profiles do. Since the baryonic cooling effect is stronger in galaxies than clusters, galactic haloes are more concentrated. In light of this, a new scenario of two halo population model is explored where galaxies are modeled as a singular isothermal sphere profile and clusters as a Navarro, Frenk and White (NFW) profile. We find the transition mass between the two has modest effects on the lensing probability. The cosmological parameter $\sigma_8$ alters the abundance of haloes and therefore affects our results. Compared with other methods, our model is simpler and more realistic. The conclusions of previous works is confirm that gravitational lensing is a natural explanation for the number count excess at the bright end., Comment: 10 pages, 10 figures, accepted by MNRAS

Cosmological defects result from cosmological phase transitions in the early Universe and the dynamics reflects their symmetry-breaking mechanisms. These cosmological defects may be probed through weak lensing effects because they interact with ordinary matters only through the gravitational force. In this paper, we investigate global textures by using weak lensing curl and B modes. Non-topological textures are modeled by the non-linear sigma model (NLSM), and induce not only the scalar perturbation but also vector and tensor perturbations in the primordial plasma due to the nonlinearity in the anisotropic stress of scalar fields. We show angular power spectra of curl and B modes from both vector and tensor modes based on the NLSM. Furthermore, we give the analytic estimations for curl and B mode power spectra. The amplitude of weak lensing signals depends on a combined parameter $\epsilon^{2}_{v} = N^{-1}\left( v/m_{\rm pl} \right)^{4}$ where $N$ and $v$ are the number of the scalar fields and the vacuum expectation value, respectively. We discuss the detectability of the curl and B modes with several observation specifications. In the case of the CMB lensing observation without including the instrumental noise, we can reach $\epsilon_{v} \approx 2.7\times 10^{-6}$. This constraint is about 10 times stronger than the current one determined from the Planck. For the cosmic shear observation, we find that the signal-to-noise ratio depends on the mean redshift and the observing number of galaxies as $\propto z^{0.7}_{\rm m}$ and $\propto N^{0.2}_{\rm g}$, respectively. In the study of textures using cosmic shear observations, the mean redshift would be one of the key design parameters., Comment: 12 pages, 4 figures, Accepted for publication in PRD

An investigation of the strong gravitational lensing in specific model of modified gravity is presented. It is shown that a slight modification to the general relativistic metric near the neighborhood of complete capture can be obtained. The formulae for the position of the critical curves, relativistic images and their magnification in the new metric are calculated and the differences with general relativity are compared. Furthermore, the radius of stable orbits is derived and the critical impact parameter is calculated. Finally, it is shown that in the strong field limit this solution may give a considerable contribution to gravitational lensing.

Correlations of galaxy ellipticities with large-scale structure, due to galactic tidal interactions, provide a potentially significant contaminant to measurements of cosmic shear. However, these intrinsic alignments are still poorly understood for galaxies at the redshifts typically used in cosmic shear analyses. For spiral galaxies, it is thought that tidal torquing is significant in determining alignments resulting in zero correlation between the intrinsic ellipticity and the gravitational potential in linear theory. Here, we calculate the leading-order correction to this result in the tidal-torque model from non-linear evolution, using second-order perturbation theory, and relate this to the contamination from intrinsic alignments to the recently-measured cross-correlation between galaxy ellipticities and the CMB lensing potential. On the scales relevant for CMB lensing observations, the squeezed limit of the gravitational bispectrum dominates the correlation. Physically, the large-scale mode that sources CMB lensing modulates the small-scale power and hence the intrinsic ellipticity, due to non-linear evolution. We find that the angular cross-correlation from tidal torquing has a very similar scale dependence as in the linear alignment model, believed to be appropriate for elliptical galaxies. The amplitude of the cross-correlation is predicted to depend strongly on the formation redshift, being smaller for galaxies that formed at higher redshift when the bispectrum of the gravitational potential was smaller. Finally, we make simple forecasts for constraints on intrinsic alignments from the correlation of forthcoming cosmic shear measurements with current CMB lensing measurements. We note that cosmic variance can be significantly reduced in measurements of the difference in the intrinsic alignments for elliptical and spiral galaxies if these can be separated (e.g., using colour)., Royal Society of New Zealand–Rutherford Foundation Trust; Cambridge Commonwealth Trust., This is the final version of the article. It first appeared from Oxford University Press via http://dx.doi.org/10.1093/mnras/stw1645

The rapidly improving precision of measurements of gravitational lensing of the Cosmic Microwave Background (CMB) also requires a corresponding increase in the precision of theoretical modeling. A commonly made approximation is to model the CMB deflection angle or lensing potential as a Gaussian random field. In this paper, however, we analytically quantify the influence of the non-Gaussianity of large-scale structure lenses, arising from nonlinear structure formation, on CMB lensing measurements. In particular, evaluating the impact of the non-zero bispectrum of large-scale structure on the relevant CMB four-point correlation functions, we find that there is a bias to estimates of the CMB lensing power spectrum. For temperature-based lensing reconstruction with CMB Stage-III and Stage-IV experiments, we find that this lensing power spectrum bias is negative and is of order one percent of the signal. This corresponds to a shift of multiple standard deviations for these upcoming experiments. We caution, however, that our numerical calculation only evaluates two of the largest bias terms and thus only provides an approximate estimate of the full bias. We conclude that further investigation into lensing biases from nonlinear structure formation is required and that these biases should be accounted for in future lensing analyses., 15+19 pages, 9 figures. Comments welcome