Lensed CMB temperature and polarization maps from the Millennium Simulation

We have constructed the first all-sky cosmic microwave background (CMB) temperature and polarization lensed maps based on a high-resolution cosmological N -body simulation, the Millennium Simulation (MS). We have exploited the lensing potential map obtained using a previously developed map-making procedure which integrates along the line-of-sight the MS dark matter distribution by stacking and randomizing the simulation boxes up to   z = 127  , and which semi-analytically supplies the large-scale power in the angular lensing potential that is not correctly sampled by the N -body simulation. The lensed sky has been obtained by properly modifying the latest version of the LensPix code to account for the MS structures. We have also produced all-sky lensed maps of the so-called  ψ _E   and  ψ _B   potentials, which are directly related to the electric and magnetic types of polarization. The angular power spectra of the simulated lensed temperature and polarization maps agree well with semi-analytic estimates up to   l ≤ 2500  , while on smaller scales we find a slight excess of power which we interpret as being due to non-linear clustering in the MS. We also observe how non-linear lensing power in the polarized CMB is transferred to large angular scales by suitably misaligned modes in the CMB and the lensing potential. This work is relevant in view of the future CMB probes, as a way to analyse the lensed sky and disentangle the contribution from primordial gravitational waves.     

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.     

Full-sky maps for gravitational lensing of the cosmic microwave background

We use the large cosmological Millennium Simulation (MS) to construct the first all-sky maps of the lensing potential and the angle, aiming at gravitational lensing of the cosmic microwave background (CMB), with the goal of properly including small-scale non-linearities and non-Gaussianity. Exploiting the Born approximation, we implement a map-making procedure based on direct ray tracing through the gravitational potential of the MS. We stack the simulation box in redshift shells up to z ∼ 11, producing continuous all-sky maps with arcmin angular resolution. A randomization scheme avoids the repetition of structures along the line of sight, and structures larger than the MS box size are added to supply the missing contribution of large-scale (LS) structures to the lensing signal. The angular power spectra of the projected lensing potential and the deflection-angle modulus agree quite well with semi-analytic estimates on scales down to a few arcmin, while we find a slight excess of power on small scales, which we interpret as being due to non-linear clustering in the MS. Our map-making procedure, combined with the LS adding technique, is ideally suited for studying lensing of CMB anisotropies, for analysing cross-correlations with foreground structures, or other secondary CMB anisotropies such as the Rees–Sciama effect.     

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.     

Precision of diffuse 21-cm lensing

We study the limits of accuracy for weak lensing maps of dark matter using diffuse 21-cm radiation from the pre-reionization epoch using simulations. We improve on previous 'optimal' quadratic lensing estimators by using shear and convergence instead of deflection angles. This is a generalization of the deflection estimator, and is more optimal for non-Gaussian sources. The cross-power spectrum of shear and convergence is an unbiased estimator of lensing power spectrum which does not require knowledge of the source four-point function. We find that non-Gaussianity provides a limit to the accuracy of weak lensing reconstruction, even if instrumental noise is reduced to zero. The best reconstruction result is equivalent to Gaussian sources with effective independent cell of side length  2.0  h ⁻¹ Mpc  . Using a source full map from z = 10 to 20, this limiting sensitivity allows mapping of dark matter at a signal-to-noise ratio greater than 1 out to l ≲ 6000, which is better than any other proposed technique for large-area weak lensing mapping.     

Large-scale cosmic microwave background anisotropies and dark energy

In this paper we investigate the effects of perturbations in a dark energy component with a constant equation of state on large-scale cosmic microwave background (CMB) anisotropies. The inclusion of perturbations increases the large-scale power. We investigate more speculative dark energy models with   w < −1  and find the opposite behaviour. Overall the inclusion of perturbations in the dark energy component increases the degeneracies. We generalize the parametrization of the dark energy fluctuations to allow for an arbitrary constant sound speed, and we show how constraints from CMB experiments change if this is included. Combining CMB with large-scale structure, Hubble parameter and supernovae observations we obtain   w =−1.02 ± 0.16 (1σ)  as a constraint on the equation of state, which is almost independent of the sound speed chosen. With the presented analysis we find no significant constraint on the constant speed of sound of the dark energy component.     

Gravitational Lensing Effect on the Two-Point Correlation of Hot Spots in the Cosmic Microwave Background

We investigate the weak gravitational lensing effect that is due to the large-scale structure of the universe on two-point correlations of local maxima (hot spots) in the two-dimensional sky map of the cosmic microwave background (CMB) anisotropy. According to the Gaussian random statistics, as most inflationary scenarios predict, the hot spots are discretely distributed, with some characteristic angular separations on the last scattering surface that are due to oscillations of the CMB angular power spectrum. The weak lensing then causes pairs of hot spots, which are separated with the characteristic scale, to be observed with various separations. We found that the lensing fairly smooths out the oscillatory features of the two-point correlation function of hot spots. This indicates that the hot spot correlations can be a new statistical tool for measuring the shape and normalization of the power spectrum of matter fluctuations from the lensing signatures.     

Weak gravitational flexion

Flexion is the significant third-order weak gravitational lensing effect responsible for the weakly skewed and arc-like appearance of lensed galaxies. Here we demonstrate how flexion measurements can be used to measure galaxy halo density profiles and large-scale structure on non-linear scales, via galaxy–galaxy lensing, dark matter mapping and cosmic flexion correlation functions. We describe the origin of gravitational flexion, and discuss its four components, two of which are first described here. We also introduce an efficient complex formalism for all orders of lensing distortion. We proceed to examine the flexion predictions for galaxy–galaxy lensing, examining isothermal sphere and Navarro–Frenk–White (NFW) profiles and both circularly symmetric and elliptical cases. We show that in combination with shear we can precisely measure galaxy masses and NFW halo concentrations. We also show how flexion measurements can be used to reconstruct mass maps in two-dimensional projection on the sky, and in three dimensions in combination with redshift data. Finally, we examine the predictions for cosmic flexion, including convergence–flexion cross-correlations, and we find that the signal is an effective probe of structure on non-linear scales.     

Numerical study of the statistical properties of the lensing excursion angles

We present ray tracing simulations combined with sets of large N -body simulations. Experiments were performed to explore, for the first time, the statistical properties of fluctuations in angular separation of nearby light-ray pairs (the so-called lensing excursion angle) induced by weak lensing by large-scale structures. We found that the probability distribution function (PDF) of the lensing excursion angles is not simply Gaussian, but has an exponential tail. It is found, however, that the tail, or more generally the non-Gaussian nature of the PDF has no significant impact on the weak lensing of the cosmic microwave background (CMB). Moreover, we found that the variance in the lensing excursion angles predicted by the power spectrum approach is in good agreement with our numerical results. These results demonstrate the validity of using the power spectrum approach to compute lensing effects on the CMB.     

lemomaf: Lensed Mock Map Facility

We present the Lensed Mock Map Facility ( lemomaf ), a tool designed to perform mock weak-lensing measurements on numerically simulated chunks of the Universe. Coupling N -body simulations to a semi-analytical model of galaxy formation, lemomaf can create realistic lensed images and mock catalogues of galaxies, at wavelengths ranging from the ultraviolet to the submillimetre. To demonstrate the power of such a tool, we compute predictions of the source–lens clustering (SLC) effect on the convergence statistics, and quantify the impact of weak lensing on galaxy counts in two different filters. We find that the SLC effect skews the probability density function of the convergence towards low values, with an intensity which strongly depends on the redshift distribution of galaxies. On the other hand, the degree of enhancement or depletion in galaxy counts due to weak lensing is independent of the SLC effect. We discuss the impact on the two-point shear statistics to be measured by future missions like SNAP and LSST . The SLC effect would bias the estimation of σ₈ from two-point statistics up to 5 per cent for a narrow redshift distribution of mean   z ∼ 0.5  , and up to 2 per cent in small angular scales for a redshift distribution of mean   z ∼ 1.5  . We conclude that accurate photometric redshifts for individual galaxies are necessary in order to quantify and isolate the SLC effect.     

An upper limit to the density inhomogeneity in the universe

An upper limit to the amplitude of the overall density fluctuation has been found by means of the gravitational lensing effect of the density inhomogeneity on the luminosities of quasars with larger redshifts. The observed differences of luminosities of quasars located at different directions are partially given by the lensing effect, therefore, a useful upper limit to the inhomogeneity can be derived if the luminosity distribution of quasars is uniform enough. We obtain that, in the case of the density parameter of the Universe =1, the overall matter should be less clustered than the luminous matter by a factor of at least 3. This result may not favour the biased dark matter scenario for the formation of large-scale structure in the Universe.     

Modelling the first probable two-plane lens system B2114+022: reproducing two compact radio cores A and D

We test possible lensing scenarios of the JVAS system B2114+022, in which two galaxies at different redshifts ('G1' at z ₁=0.3157 and 'G2' at z ₂=0.5883) are found within 2 arcsec of quadruple radio sources. For our investigation, we use possible lensing constraints derived from a wealth of data on the radio sources obtained with VLA, MERLIN, VLBA and EVN as well as HST imaging data on the two galaxies, which were presented recently in Augusto et al. In the present study, we focus on reproducing the widest separated, observationally similar radio components A and D as lensed images. We first treat G2 (which is the more distant one from the geometric centre) as a shear term, and then consider two-plane lensing explicitly including G2's potential at the z ₂ plane as the first case of two-plane lens modelling. Our modelling results not only support the hypothesis that the system includes gravitationally lensed images of a higher-redshift extragalactic object, but they also show that the explicit inclusion of G2's potential at the second lens plane is necessary in order to fit the data with astrophysically plausible galaxy parameters. Finally, we illustrate a natural consequence of a two-plane lens system, namely the prediction of distortion as well as shift and stretching of G2's isophotes by G1's potential, which can in principle be measured by subtracting out G1's light distribution in an image of high signal-to-noise ratio and good angular resolution, especially a multicolour one.     

The dark matter environment of the Abell 901/902 supercluster: a weak lensing analysis of the HST STAGES survey

We present a high-resolution dark matter reconstruction of the   z = 0.165  Abell 901/902 supercluster from a weak lensing analysis of the Hubble Space Telescope STAGES survey. We detect the four main structures of the supercluster at high significance, resolving substructure within and between the clusters. We find that the distribution of dark matter is well traced by the cluster galaxies, with the brightest cluster galaxies marking out the strongest peaks in the dark matter distribution. We also find a significant extension of the dark matter distribution of Abell 901a in the direction of an infalling X-ray group Abell 901α. We present mass, mass-to-light and mass-to-stellar mass ratio measurements of the structures and substructures that we detect. We find no evidence for variation of the mass-to-light and mass-to-stellar mass ratio between the different clusters. We compare our space-based lensing analysis with an earlier ground-based lensing analysis of the supercluster to demonstrate the importance of space-based imaging for future weak lensing dark matter 'observations'.     

Improving efficiency in radio surveys for gravitational lenses

Many lens surveys have hitherto used observations of large samples of background sources to select the small minority which are multiply imaged by lensing galaxies along the line of sight. Recently surveys such as SLACS and OLS have improved the efficiency of surveys by pre-selecting double-redshift systems from SDSS. We explore other ways to improve survey efficiency by optimum use of astrometric and morphological information in existing large-scale optical and radio surveys. The method exploits the small position differences between FIRST radio positions of lensed images and the SDSS lens galaxy positions, together with the marginal resolution of some larger gravitational lens systems by the FIRST beam. We present results of a small pilot study with the VLA and MERLIN, and discuss the desirable criteria for future surveys.     

B0712+472: a new radio four-image gravitational lens

A new four-image gravitational lens system, B0712+472, has been discovered during the Cosmic Lens All-Sky Survey. This system consists of four flat-spectrum radio images that are also seen on a Hubble Space Telescope ( HST ) image, together with the lensing galaxy. We present MERLIN, VLA and VLBA maps and WHT spectra of the system as well as the HST images. The light distribution of the lensing galaxy is highly elongated and so too is the mass distribution deduced from modelling. We suggest a redshift of ∼1.33 for the lensed object; the lens redshift will require further investigation. The discovery of this new system further increases the ratio of four-image to two-image lens systems currently known, exacerbating problems of required ellipticity of matter distributions in lensing galaxies.     

Galaxy density profiles and shapes – I. Simulation pipeline for lensing by realistic galaxy models

Studies of strong gravitational lensing in current and upcoming wide and deep photometric surveys, and of stellar kinematics from (integral-field) spectroscopy at increasing redshifts, promise to provide valuable constraints on galaxy density profiles and shapes. However, both methods are affected by various selection and modelling biases, which we aim to investigate in a consistent way. In this first paper in a series, we develop a flexible but efficient pipeline to simulate lensing by realistic galaxy models. These galaxy models have separate stellar and dark matter components, each with a range of density profiles and shapes representative of early-type, central galaxies without significant contributions from other nearby galaxies. We use Fourier methods to calculate the lensing properties of galaxies with arbitrary surface density distributions, and Monte Carlo methods to compute lensing statistics such as point-source lensing cross-sections. Incorporating a variety of magnification bias modes lets us examine different survey limitations in image resolution and flux. We rigorously test the numerical methods for systematic errors and sensitivity to basic assumptions. We also determine the minimum number of viewing angles that must be sampled in order to recover accurate orientation-averaged lensing quantities. We find that for a range of non-isothermal stellar and dark matter density profiles typical of elliptical galaxies, the combined density profile and corresponding lensing properties are surprisingly close to isothermal around the Einstein radius. The converse implication is that constraints from strong lensing and/or stellar kinematics, which are indeed consistent with isothermal models near the Einstein radius, cannot trivially be extrapolated to smaller and larger radii.     

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.     

CLASS B2108+213: a new wide-separation gravitational lens system

We present observations of CLASS B2108+213, the widest separation gravitational lens system discovered by the Cosmic Lens All-Sky Survey. Radio imaging using the VLA at 8.46 GHz and MERLIN at 5 GHz shows two compact components separated by 4.56 arcsec with a faint third component in between which we believe is emission from a lensing galaxy. 5-GHz VLBA observations reveal milliarcsecond-scale structure in the two lensed images that is consistent with gravitational lensing. Optical emission from the two lensed images and two lensing galaxies within the Einstein radius is detected in Hubble Space Telescope imaging. Furthermore, an optical gravitational arc, associated with the strongest lensed component, has been detected. Surrounding the system is a number of faint galaxies which may help explain the wide image separation. A plausible mass distribution model for CLASS B2108+213 is also presented.     

Gravitational lensing of extended high-redshift sources by dark matter haloes

High-redshift galaxies and quasi-stellar objects (QSOs) are most likely to be strongly lensed by intervening haloes between the source and the observer. In addition, a large fraction of lensed sources is expected to be seen in the submillimetre region, as a result of the enhanced magnification bias on the steep intrinsic number counts. We extend in three directions Blain's earlier study of this effect.
First, we use a modification of the Press–Schechter mass function and detailed lens models to compute the magnification probability distribution. We compare the magnification cross-sections of populations of singular isothermal spheres and Navarro, Frenk & White (NFW) haloes and find that they are very similar, in contrast to the image-splitting statistics which were recently investigated in other studies. The distinction between the two types of density profile is therefore irrelevant for our purposes.
Secondly, we discuss quantitatively the maximum magnification, μ _max, that can be achieved for extended sources (galaxies) with realistic luminosity profiles, taking into account the possible ellipticity of the lensing potential. We find that μ _max plausibly falls into the range     for sources of     effective radius at redshifts within     .
Thirdly, we apply our model for the lensing magnification to a class of sources following the luminosity evolution typical for a unified scheme of QSO formation. As a result of the peculiar steepness of their intrinsic number counts, we find that the lensed source counts at a fiducial wave length of 850 μm can exceed the unlensed counts by several orders of magnitude at flux densities ≳100 mJy, even with a conservative choice of the maximum magnification.