Analytic relations between the observed gravitational microlensing parameters with and without the effect of blending

When a microlensing light curve is contaminated by blended light from unresolved stars near the line of sight to the lensed star, the light curve shape and corresponding parametrization for the event will differ from the values expected when the event is not affected by blending. As a result, blending makes it difficult to identify the major lens population and to estimate the amount of lensing matter. In order to estimate the effect of blending on the result of lensing experiments, it is, therefore, essential to know how the observed lensing parameters change depending on the fraction of blended light. Previously, the changed lensing parameters were obtained with a statistical method that not only required a large amount of computation time but also was prone to uncertainty. In this paper, we derive analytic relations between the lensing parameters with and without the effect of blending. By using these relations, we investigate the dependence of the observed lensing parameters on the amount of blended light, the impact parameter and the threshold amplification for event detection.     

Gravitational lensing in eclipsing binary stars

I consider the effect of the gravitational deflection of light upon the light curves of eclipsing binary stars, focusing mainly upon systems containing at least one white dwarf component. In absolute terms the effects are small, however they are strongest at the time of secondary eclipse when the white dwarf transits its companion, and act to reduce the depth of this feature. If not accounted for, this may lead to under-estimation of the radius of the white dwarf compared with that of its companion. I show that the effect is significant for plausible binary parameters, and that it leads to ∼25 per cent reduction in the transit depth in the system KPD 1930+2752. The reduction of eclipse depth is degenerate with the stellar radius ratio, and therefore cannot be used to establish the existence of lensing. A second-order effect of the light bending is to steepen the ingress and egress features of the secondary eclipse relative to the primary eclipse, although it will be difficult to see this in practice. I consider also binaries containing neutron stars and black holes. I conclude that, although relatively large effects are possible in such systems, a combination of rarity, faintness and intrinsic variability makes it unlikely that lensing will be detectable in them.     

On the astrometric behaviour of binary microlensing events

Despite the suspected binarity for a significant fraction of Galactic lenses, the current photometric surveys detected binary microlensing events only for a small fraction of the total events. The detection efficiency is especially low for non-caustic crossing events, which comprise the majority of the binary lensing events, as a result of the absence of distinctive features in their light curves combined with small deviations from the standard light curve of a single point-mass event. In addition, even if they are detected, it will be difficult to determine the solution of the binary lens parameters owing to the severe degeneracy problem. In this paper, we investigate the properties of binary lensing event expected when they are astrometrically observed by using high-precision interferometers. For this, we construct vector field maps of excess centroid shifts, which represent the deviations of the binary lensing centroid shifts from those of a single lensing event as a function of source position. From the analysis of the maps, we find that the excess centroid shifts are substantial in a considerably large area around caustics. In addition, they have characteristic sizes and directions depending strongly on the source positions with respect to the caustics and the resulting trajectories of the light centroid (astrometric trajectories) have distinctive features, which can be distinguished from the deviations caused by other reasons. We classify the types of the deviations and investigate where they occur. Because of the strong dependence of the centroid shifts on the lens system geometry combined with the distinctive features in the observed astrometric trajectories, astrometric binary lensing observations will provide an important tool that can probe the properties of the Galactic binary lens population.     

Gravitational lens under perturbations: symmetry of perturbing potentials with invariant caustics

When the gravitational lensing potential can be approximated by that of a circularly symmetric system affected by weak perturbations, it is found that the shape of the resulting (tangential) caustics is entirely specified by the local azimuthal behaviour of the affecting perturbations. This provides a common mathematical groundwork for understanding problems such as the close–wide  ( d ↔ d ⁻¹)  separation degeneracy of binary lens microlensing light curves and the shear–ellipticity degeneracy of quadruple image lens modelling.     

Variable radio echoes from a jet in QSO 0957+561

Time delay determinations in astrophysics are used most often to find time delays between flux density variations of different spectral bands and/or spectral lines in AGNs and different images of gravitationally lensed QSOs. Here we consider a new algorithm for a complex case, when the time delay is itself a linear function of time and the intensity of echo response is power function of the delay. We apply this method to investigate optical-to-radio delay in the double quasar 0957+561, which is a generally accepted case of gravitational lensing.Radio-optical correlation in QSO 0957+561 was first reported by Oknyanskij and Beskin (1993, hereafter OB) on the basis of radio observations made in the years 1979 to 1990. OB used an idea to take into account the known gravitational lensing time delay to get combined radio and optical light curves and then to use them for determination of the possible radio-from-optical time delay. It was found this way that radio variations (5 MHz) followed optical ones by about 6.4 years with high level of correlation (0.87). Using new radio data (Haarsmaet al., 1996), for the interval 1979–1994 we find nearly the same value for the optical-to-radio delay as has been found before. Additionally we suspect that the time delay value is linearly increasing at about 110 days per year while the portion of reradiated flux in the radioresponse is decreasing.Obtained results indicate that the optical and radio emitting regions are physically related, but have distinct size scales, locations and possibly radiation mechanisms. We conclude that the results can be explained by simple model were the variable radio source is ejected from the central part of the QSO compact component and that the changing time delay between the optical and radio light curves is consequence of light travel effect.     

Kinematic effect in gravitational lensing by clusters of galaxies

Gravitational lensing provides an efficient tool for the investigation of matter structures, independent of the dynamical or the hydrostatic equilibrium properties of the deflecting system. However, it depends on the kinematic status. In fact, either a translational motion or a coherent rotation of the mass distribution can affect the lensing properties. Here, light deflection by galaxy clusters in motion is considered. Even if gravitational lensing mass measurements of galaxy clusters are regarded as very reliable estimates, the kinematic effect should be considered. A typical peculiar motion with respect to the Hubble flow brings about a systematic error ≲0.3 per cent, independent of the mass of the cluster. On the other hand, the effect of the spin increases with the total mass. For cluster masses  ∼10¹⁵ M_⊙  , the effect of the gravitomagnetic term is ≲0.04 per cent on strong lensing estimates and ≲0.5 per cent in the weak-lensing analyses. The total kinematic effect on the mass estimate is then ≲1 per cent, which is negligible in current statistical studies. In the weak-lensing regime, the rotation imprints a typical angular modulation in the tangential shear distortion. This would allow, in principle, a detection of the gravitomagnetic field and a direct measurement of the angular velocity of the cluster but the required background source densities are well beyond current technological capabilities.     

Weak lensing in the second post-Newtonian approximation: gravitomagnetic potentials and the integrated Sachs–Wolfe effect

Dark matter currents in the large-scale structure give rise to gravitomagnetic terms in the metric, which affect the light propagation. Corrections to the weak-lensing power spectrum due to these gravitomagnetic potentials are evaluated by perturbation theory. A connection between gravitomagnetic lensing and the integrated Sachs–Wolfe (iSW) effect is drawn, which can be described by a line-of-sight integration over the divergence of the gravitomagnetic vector potential. This allows the power spectrum of the iSW-effect to be derived within the framework of the same formalism as derived for gravitomagnetic lensing and reduces the iSW-effect to a second-order lensing phenomenon. The three-dimensional power spectra are projected by means of a generalized Limber-equation to yield the angular power spectra. Gravitomagnetic corrections to the weak-lensing spectrum are negligible at currently accessible scales, and cosmic-variance considerations suggest that the detection of the iSW-effect's contribution to the cosmic microwave background angular power spectrum is too small to be detectable at multipoles probed by the Planck satellite.     

Rotation in gravitational lenses

Gravitational lensing deflects light. A single lens deflector can only shear images, but cannot induce rotations. Multiple lens planes can induce rotations. Such rotations can be observed in quadruply imaged sources, and can be used to distinguish between two proposed solutions of the flux anomaly problem: substructures in lensing galaxies versus large-scale structure. We predict the expected amount of rotation due to large-scale structure in strong lensing systems, and show how this effect can be measured using ∼mas very long baseline interferometry astrometry of quadruple lenses with extended source structures. The magnitude of rotation is around 1°. The biggest theoretical uncertainty is the power spectrum of dark matter on very small scales. This procedure can potentially be turned around to measure the dark matter power spectrum on very small scales. We list the predicted rms rotation angles for several quadruple lenses with known lens and source redshifts.     

Empirical constraints on alternative gravity theories from gravitational lensing

If it is hypothesized that there is no dark matter, then some alternative gravitational theory must take the place of general relativity (GR) on the largest scales. Dynamical measurements can be used to investigate the nature of such a theory, but only where there is visible matter. Gravitational lensing is potentially a more powerful probe as it can be used to measure deflections far from the lens and, for sufficiently large separations, allow it to be treated as a point-mass. Microlensing within the local group does not yet provide any interesting constraints, as only images formed close to the deflectors are appreciably magnified, but stacking of multiple light-curves and observations of microlensing on cosmological scales may be able to discriminate between GR and non-dark matter theories. Galaxy–galaxy lensing is likely to be a more powerful probe of gravity, with the Sloan Digital Sky Survey (SDSS) commissioning data used here to constrain the deflection law of galaxies to be     for impact parameters in the range     . Together with observations of flat rotation curves, these results imply that, in any gravitational theory, photons must experience (close to) twice the deflection of massive particles moving at the speed of light (at least on these physical scales). The full SDSS data set will also be sensitive to asymmetry in the lensing signal and to variation of the deflection law with galaxy type. A detection of either of these effects would represent an independent confirmation that galaxies are dark matter-dominated; conversely, azimuthal symmetry of the shear signal would rule out the typically ellipsoidal haloes predicted by most simulations of structure formation.     

Intrinsic correlation of galaxy shapes: implications for weak lensing measurements

Weak gravitational lensing is now established as a powerful method to measure mass fluctuations in the universe. It relies on the measurement of small coherent distortions of the images of background galaxies. Even low-level correlations in the intrinsic shapes of galaxies could however produce a significant spurious lensing signal. These correlations are also interesting in their own right, since their detection would constrain models of galaxy formation. Using     haloes found in N -body simulations, we compute the correlation functions of the intrinsic ellipticity of spiral galaxies assuming that the disc is perpendicular to the angular momentum of the dark matter halo. We also consider a simple model for elliptical galaxies, in which the shape of the dark matter halo is assumed to be the same as that of the light. For deep lensing surveys with median redshifts ∼1, we find that intrinsic correlations of ∼10⁻⁴ on angular scales     are generally below the expected lensing signal, and contribute only a small fraction of the excess signals reported on these scales. On larger scales we find limits to the intrinsic correlation function at a level ∼10⁻⁵, which gives a (model-dependent) range of separations for which the intrinsic signal is about an order of magnitude below the ellipticity correlation function expected from weak lensing. Intrinsic correlations are thus negligible on these scales for dedicated weak lensing surveys. For wider but shallower surveys such as SuperCOSMOS, APM and SDSS, we cannot exclude the possibility that intrinsic correlations could dominate the lensing signal. We discuss how such surveys could be used to calibrate the importance of this effect, as well as study spin–spin correlations of spiral galaxies.     

Strong lensing optical depths in a ΛCDM universe

We investigate strong gravitational lensing in the concordance ΛCDM cosmology by carrying out ray tracing along past light cones through the Millennium Simulation, the largest simulation of cosmic structure formation ever carried out. We extend previous ray-tracing methods in order to take full advantage of the large volume and the excellent spatial and mass resolution of the simulation. As a function of source redshift we evaluate the probability that an image will be highly magnified, will be highly elongated or will be one of a set of multiple images. We show that such strong lensing events can almost always be traced to a single dominant lensing object and we study the mass and redshift distribution of these primary lenses. We fit analytic models to the simulated dark haloes in order to study how our optical depth measurements are affected by the limited resolution of the simulation and of the lensing planes that we construct from it. We conclude that such effects lead us to underestimate total strong lensing cross-sections by about 15 per cent. This is smaller than the effects expected from our neglect of the baryonic components of galaxies. Finally we investigate whether strong lensing is enhanced by material in front of or behind the primary lens. Although strong lensing lines of sight are indeed biased towards higher than average mean densities, this additional matter typically contributes only a few per cent of the total surface density.     

Strong gravitational lensing in a charged squashed Kaluza-Klein black hole

In this paper we investigate the strong gravitational lensing in a charged squashed Kaluza-Klein black hole. We suppose that the supermassive black hole in the galaxy center can be considered by a charged squashed Kaluza-Klein black hole and then we study the strong gravitational lensing theory and estimate the numerical values for parameters and observables of it. We explore the effects of the scale of extra dimension ρ ₀ and the charge of black hole ρ _q on these parameters and observables.     

On Einstein clusters as galactic dark matter haloes

We consider global and gravitational lensing properties of the recently suggested Einstein clusters of weakly interacting massive particles (WIMPs) as galactic dark matter haloes. Being tangential pressure dominated, Einstein clusters are strongly anisotropic systems which can describe any galactic rotation curve by specifying the anisotropy. Due to this property, Einstein clusters may be considered as dark matter candidates. We analyse the stability of the Einstein clusters against both radial and non-radial pulsations, and we show that the Einstein clusters are dynamically stable. With the use of the Buchdahl type inequalities for anisotropic bodies, we derive upper limits on the velocity of the particles defining the cluster. These limits are consistent with those obtained from stability considerations. The study of light deflection shows that the gravitational lensing effect is slightly smaller for the Einstein clusters as compared to the singular isothermal density sphere model for dark matter. Therefore, lensing observations may discriminate, at least, in principle, between Einstein cluster and the other dark matter models.     

Statistical distribution of gravitational-lensing excursion angles: winding ways to us from the deep Universe

We investigate statistical distributions of differences in gravitational-lensing deflections between two light rays, the so-called lensing excursion angles. A probability distribution function of the lensing excursion angles, which plays a key role in estimates of lensing effects on angular clustering of objects (such as galaxies, quasi-stellar objects and also the cosmic microwave background temperature map), is known to consist of two components: a Gaussian core and an exponential tail. We use numerical gravitational-lensing experiments in a ΛCDM cosmology for quantifying these two components. We especially focus on the physical processes responsible for generating these two components. We develop a simple empirical model for the exponential tail which allows us to explore its origin. We find that the tail is generated by the coherent lensing scatter by massive haloes with   M > 10¹⁴  h ⁻¹ M_⊙  at   z < 1  and that its exponential shape arises due to the exponential cut-off of the halo mass function at that mass range. On scales larger than 1 arcmin, the tail does not have a practical influence on the lensing effects on the angular clustering. Our model predicts that the coherent scatter may have non-negligible effects on angular clustering at subarcminute scales.     

Gravitational lensing of anisotropic sources

In strong gravitational lensing, the multiple images we see correspond to light rays that leave the source in slightly different directions. If the source emission is anisotropic, the images may differ from conventional lensing predictions (which assume isotropy). To identify scales on which source anisotropy may be important, we study the angle δ between the light rays emerging from the source, for different lensing configurations. If the lens has a power-law profile   M ∝ R ^γ  , the angle δ initially increases with lens redshift and then either diverges (for a steep profile  γ < 1  ), remains constant (for an isothermal profile  γ= 1  ), or vanishes (for a shallow profile  γ > 1  ) as   z _l→ z _s  . The scaling with lens mass is roughly  δ∝ M ^1/(2−γ)  . The results for an Navarro–Frenk–White (NFW) profile are qualitatively similar to those for a shallow power law, with δ peaking at about half the redshift of the source (not half the distance). In practice, beaming could modify the statistics of beamed sources lensed by massive clusters: for an opening angle  θ_jet  , there is a probability as high as   P ∼ 0.02–0.07(θ_jet/0.5°)⁻¹  that one of the lensed images may be missed (for  2 ≲ z _s≲ 6  ). Differential absorption within active galactic nuclei (AGNs) could modify the flux ratios of AGNs lensed by clusters; a sample of AGNs lensed by clusters could provide further constraints on the sizes of absorbing regions. Source anisotropy is not likely to be a significant effect in galaxy-scale strong lensing.     

Q2237+0305 source structure and dimensions from light-curve simulation

Assuming a two-component quasar structure model consisting of a central compact source and an extended outer feature, we produce microlensing simulations for a population of compact masses in the lensing galaxy of Q2237+0305. Such a model is a simplified version of that adopted to explain the brightness variations observed in Q0957. The microlensing light curves generated for a range of source parameters were compared to the light curves obtained in the framework of the Optical Gravitational Lensing Experiment program. With a large number of trials, we built, in the domain of the source structure parameters, probability distributions to find 'good' realizations of light curves. The values of the source parameters which provide the maximum of the joint probability distribution calculated for all the image components have been accepted as estimates for the source structure parameters. The results favour the two-component model of the quasar brightness structure over a single compact central source model, and in general the simulations confirm the Schild–Vakulik model that previously described successfully the microlensing and other properties of Q0957. Adopting 3300 km s⁻¹ for the transverse velocity of the source, the effective size of the central source was determined to be about  2 × 10¹⁵ cm  , and  ɛ≈ 2  was obtained for the ratio of the integral luminosity of the outer feature to that of the central source.     

Imaging the cosmic matter distribution using gravitational lensing of pre-galactic H i

21-cm emission from neutral hydrogen during and before the epoch of cosmic reionization is gravitationally lensed by material at all lower redshifts. Low-frequency radio observations of this emission can be used to reconstruct the projected mass distribution of foreground material, both light and dark. We compare the potential imaging capabilities of such 21-cm lensing with those of future galaxy lensing surveys. We use the Millennium Simulation to simulate large-area maps of the lensing convergence with the noise, resolution and redshift-weighting achievable with a variety of idealized observation programmes. We find that the signal-to-noise ratio of 21-cm lens maps can far exceed that of any map made using galaxy lensing. If the irreducible noise limit can be reached with a sufficiently large radio telescope, the projected convergence map provides a high-fidelity image of the true matter distribution, allowing the dark matter haloes of individual galaxies to be viewed directly, and giving a wealth of statistical and morphological information about the relative distributions of mass and light. For instrumental designs like that planned for the Square Kilometre Array, high-fidelity mass imaging may be possible near the resolution limit of the core array of the telescope.     

Weak lensing of the CMB: extraction of lensing information from the trispectrum

We discuss the four-point correlation function, or the trispectrum in Fourier space, of CMB temperature and polarization anisotropies due to the weak gravitational lensing effect by intervening large scale structure. We discuss the squared temperature power spectrum as a probe of this trispectrum and, more importantly, as an observational approach to extracting the power spectrum of the deflection angle associated with the weak gravitational lensing effect on the CMB. We extend previous discussions on the trispectrum and associated weak lensing reconstruction from CMB data by calculating non-Gaussian noise contributions, beyond the previously discussed dominant Gaussian noise. Non-Gaussian noise contributions are generated by lensing itself and by the correlation between the lensing effect and other foreground secondary anisotropies in the CMB such as the Sunyaev–Zel’dovich (SZ) effect. When the SZ effect is removed from temperature maps using its spectral dependence, we find these additional non-Gaussian noise contributions to be an order of magnitude lower than the dominant Gaussian noise. If the noise-bias due to the dominant Gaussian part of the temperature squared power spectrum is removed, then these additional non-Gaussian contributions provide the limiting noise level for the lensing reconstruction. The temperature squared power spectrum allows a high signal-to-noise extraction of the lensing deflections and a confusion-free separation of the curl (or B-mode) polarization due to inflationary gravitational waves from that due to lensed gradient (or E-mode) polarization. The small angular scale temperature and polarization anisotropy measurements provide a novel approach to weak lensing studies, complementing the approach based on galaxy ellipticities.     

Testing Bekenstein's relativistic Modified Newtonian Dynamics with lensing data

We propose to use multiple-imaged gravitational lenses to set limits on gravity theories without dark matter, specifically tensor–vector–scalar (TeVeS) theory, a theory which is consistent with fundamental relativistic principles and the phenomenology of Modified Newtonian Dynamics (MOND) theory. After setting the framework for lensing and cosmology, we analytically derive the deflection angle for the point lens and the Hernquist galaxy profile, and study their patterns in convergence, shear and amplification. Applying our analytical lensing models, we fit galaxy-quasar lenses in the CfA-Arizona Space Telescope Lens Survey (CASTLES) sample. We do this with three methods, fitting the observed Einstein ring sizes, the image positions, or the flux ratios. In all the cases, we consistently find that stars in galaxies in MOND/TeVeS provide adequate lensing. Bekenstein's toy μ function provides more efficient lensing than the standard MOND μ function. But for a handful of lenses, a good fit would require a lens mass orders of magnitude larger/smaller than the stellar mass derived from luminosity unless the modification function μ and modification scale a ₀ for the universal gravity were allowed to be very different from what spiral galaxy rotation curves normally imply. We discuss the limitation of present data and summarize constraints on the MOND μ function. We also show that the simplest TeVeS 'minimal-matter' cosmology, a baryonic universe with a cosmological constant, can fit the distance–redshift relation from the supernova data, but underpredicts the sound horizon size at the last scattering. We conclude that lensing is a promising approach to differentiate laws of gravity.     

Time resolved images from the center of the Galaxy appear to counter General Relativity

Intense observations of the galactic center since 1992 have revealed the presence of a supermassive object located there, some 26 000 light years from Earth. The mass of the galactic center was determined using time resolved astrometry over a time span of 13 years, from 1992 to present. The observations clearly show that the stars in the immediate vicinity of the supermassive galactic center, denoted as Sagittarius A* (Sgr A*), move along purely Keplerian orbits around Str A*. Observation of the rapidly moving stars permitted astrophysicists to determine a mass for the galactic center of around 3.6 million solar masses. Time resolved images of the Keplerian motions of these stars has exhibited to date no evidence of distortions in the images due to gravitational light bending effects, as predicted by General Relativity. In this paper, a well known tool commonly used by astrophysicists for estimating the effect of gravitation on light rays was examined. The results reveal flaws in the understanding of fundamental principles in mathematical physics applied to gravitational effects on rays of light, as predicted by General Relativity, at the site of a point‐like gravitating masses such as the galactic center mass. Application of the Gauss Law to point‐like gravitating masses shows that a requirement for the colinear alignment of the light source, the lensing and the observer is not necessary for an observation of gravitational lensing as predicted by General Relativity. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)