arXiv subjects
Publications and source records attributed to Nick Kaiser.
In this introductory talk I give an elementary overview of the development, current status, and near term outlook for three probes of large scale structure; galaxy clustering, bulk flows and weak lensing observations.
A tangential distortion of background source galaxies around foreground lens galaxies in the Hubble Deep Field is detected at the 99.3% confidence level. An important element of our analysis is the use of photometric redshifts to determine distances of lens and source galaxies and rest-frame B-band luminosities of the lens galaxies. The lens galaxy halos obey a Tully-Fisher relation between halo circular velocity and luminosity; the typical lens galaxy, at a redshift z = 0.6, has a circular velocity of 210 +/-40 km/s at M_B = -18.5, if q_0 = 0.5. Control tests, in which lens and source positions and source ellipticities are randomized, confirm the significance level of the detection quoted above. Furthermore, a marginal signal is also detected from an independent, fainter sample of source galaxies without photometric redshifts. Potential systematic effects, such as contamination by aligned satellite galaxies, the distortion of source shapes by the light of the foreground galaxies, PSF anisotropies, and contributions from mass distributed on the scale of galaxy groups are shown to be negligible. A comparison of our result with the local Tully-Fisher relation indicates that intermediate-redshift galaxies are fainter than local spirals by 1.0 +/- 0.6 B mag at a fixed circular velocity. This is consistent with some spectroscopic studies of the rotation curves of intermediate-redshift galaxies. This result suggests that the strong increase in the global luminosity density with redshift is dominated by evolution in the galaxy number density.
We explore the dependence of weak lensing phenomena on the background cosmology. We first generalise the relation between $P_ψ(ω)$, the angular power spectrum of the distortion, and the power spectrum of density fluctuations to non-flat cosmologies. We then compute $P_ψ$ for various illustrative models. A useful cosmological discriminator is the growth of $P_ψ$ with source redshift which is much stronger in low matter density models, and especially in $Λ$-dominated models. With even crude redshift information (say from broad band colours) it should be possible to constrain the cosmological world model. The amplitude of $P_ψ(ω)$ is also quite sensitive to the cosmology, but requires a reliable external normalisation for the mass fluctuations. If one normalises to galaxy clustering, with $M/L$ fixed by small-scale galaxy dynamics, then low density models predict a much stronger distortion. If, however, one normalises to large-scale bulk-flows, the predicted distortion for sources at redshifts $Z_s \sim 1-3$ is rather insensitive to the background cosmology. The signals predicted here can be detected at a very high level of significance with a photometric survey covering say 10 square degrees, but sparse sampling is needed to avoid large sampling variance and we discuss the factors influencing the design of an optimum survey. Turning to weak lensing by clusters we find that for high lens redshifts ($Z_l\simeq1$) the critical density is substantially reduced in $Λ$ models, but that the ratio of the shear or convergence to the velocity dispersions or X-ray temperature of clusters is only very weakly dependent on the cosmology.
We describe the statistical properties of light rays propagating though a random sea of gravity waves and compare with the case for scalar metric perturbations from density inhomogeneities. For scalar fluctuations the deflection angle grows as the square-root of the path length $D$ in the manner of a random walk, and the rms displacement of a ray from the unperturbed trajectory grows as $D^{3/2}$. For gravity waves the situation is very different. The mean square deflection angle remains finite and is dominated by the effect of the metric fluctuations at the ends of the ray, and the mean square displacement grows only as the logarithm of the path length. In terms of power spectra, the displacement for scalar perturbations has $P(k) \propto 1/ k^4$ while for gravity waves the trajectories of photons have $P(k) \propto 1/k$ which is a scale-invariant or `flicker-noise' process, and departures from rectilinear motion are suppressed, relative to the scalar case, by a factor $\sim (λ/ D)^{3/2}$ where $λ$ is the characteristic scale of the metric fluctuations and $D$ is the path length. This result casts doubt on the viability of some recent proposals for detecting or constraining the gravity wave background by astronomical measurements.
In Paper I we studied the theory of gravitational microlensing for a planar distribution of point masses. In this second paper, we extend the analysis to a three-dimensional lens distribution. First we study the lensing properties of three-dimensional lens distributions by considering in detail the critical curves, the caustics, the illumination patterns, and the magnification cross-sections sigma(A) of multiplane configurations with two, three, and four point masses. For N_\ast point masses that are widely separated in Lagrangian space, we find that there are ~ 2^{N_\ast} - 1 critical curves in total, but that only ~ N_\ast of these produce prominent caustic-induced features at the high magnification end of sigma(A). In the case of a low optical depth random distribution of point masses, we show that the multiplane lens equation near a point mass can be reduced to the single plane equation of a point mass perturbed by weak shear. This allows us to calculate the caustic-induced feature in the macroimage magnification distribution P(A) as a weighted sum of the semi-analytic feature derived in Paper I for a planar lens distribution. The resulting semi-analytic caustic-induced feature is similar to the feature in the planar case, but it does not have any simple scaling properties, and it is shifted to higher magnification. The semi-analytic distribution is compared to the results of previous numerical simulations for optical depth tau ~ 0.1, and they are in better agreement than a similar comparison in the planar case. We explain this by estimating the fraction of caustics of individual lenses that merge with those of their neighbors. For tau=0.1, the fraction is approx 20%, much less than the approx 55% for the planar case.
(Abridged) In this paper we refine the theory of microlensing for a planar distribution of point masses. We derive the macroimage magnification distribution P(A) at high magnification (A-1 >> tau^2) for a low optical depth (tau << 1) lens distribution by modeling the illumination pattern as a superposition of the patterns due to individual ``point mass plus weak shear'' lenses. We show that a point mass plus weak shear lens produces an astroid- shaped caustic and that the magnification cross-section obeys a simple scaling property. By convolving this cross-section with the shear distribution, we obtain a caustic-induced feature in P(A) which also exhibits a simple scaling property. This feature results in a 20% enhancement in P(A) at A approx 2/tau. In the low magnification (A-1 << 1) limit, the macroimage consists of a bright primary image and a large number of faint secondary images formed close to each of the point masses. Taking into account the correlations between the primary and secondary images, we derive P(A) for low A. The low-A distribution has a peak of amplitude ~ 1/tau^2 at A-1 ~ tau^2 and matches smoothly to the high-A distribution. We combine the high- and low-A results and obtain a practical semi-analytic expression for P(A). This semi-analytic distribution is in qualitative agreement with previous numerical results, but the latter show stronger caustic-induced features at moderate A for tau as small as 0.1. We resolve this discrepancy by re-examining the criterion for low optical depth. A simple argument shows that the fraction of caustics of individual lenses that merge with those of their neighbors is approx 1-exp(-8 tau). For tau=0.1, the fraction is surprisingly high: approx 55%. For the purpose of computing P(A) in the manner we did, low optical depth corresponds to tau << 1/8.
We report the detection of weak gravitational lensing of faint, distant background galaxies by the rich, X-ray luminous cluster of galaxies MS1054-03 at z=0.83. This is the first measurement of weak lensing by a bona fide cluster at such a high redshift. We detect tangential shear at the 5% - 10% level over a range of radii 50'' < r < 250'' centered on the optical position of the cluster. Two-dimensional mass reconstruction using galaxies with 21.5 < I < 25.5 shows a strong peak which coincides with the peak of the smoothed cluster light distribution. Splitting this sample by magnitude (at I = 23.5) and color (at R-I = 0.7), we find that the brighter and redder subsamples are only very weakly distorted, indicating that the faint blue galaxies (FBG's), which dominate the shear signal, are relatively more distant. The derived cluster mass is quite sensitive to the N(z) for the FBG's. At one extreme, if all the FBG's are at z_s = 3, then the mass within a $0.5h^{-1}$Mpc aperture is $(5.9 \pm 1.24)\times 10^{14}$\h1 $M_\odot$, and the mass-to-light ratio is $M/L_V = 350 \pm 70 h$ in solar units. For $z_s = 1.5$ the derived mass is $\sim$70\% higher and $M/L \simeq 580 h$. If $N(z)$ follows the no evolution model (in shape) then $M/L \simeq 800h$, and if all the FBG's lie at $z_s\la 1$ the required $M/L$ exceeds $1600h$. These data provide clear evidence that large, dense mass concentrations existed at early epochs; that they can be weighed efficiently by weak lensing observations; and that most of the FBG's are at high redshift.
Weak lensing observations measure the shear field and hence the gradient of the dimensionless surface density $κ$. We present several new algorithms to recover $κ$ {}from shear estimates on a finite region and compare how they perform with realistically noisy data. The reconstruction methods studied here are divided into 2 classes: direct reconstruction and regularized inversion. Direct reconstruction techniques express $κ$ as a 2D integral of the shear field. This yields an estimator for $κ$ as a discrete sum over background galaxy ellipticities which is straightforward to implement, and allows a simple estimate of the noise. We study 3 types of direct reconstruction methods: 1) $κ$-estimators that measure the surface density at any given target point relative to the mean value in a reference region 2) a method that explicitly attempts to minimize the rotational part of $\nabla κ$ that is due to noise and 3) a novel, exact Fourier-space inverse gradient operator. We also develop two `regularized maximum likelihood' methods, one of which employs the conventional discrete Laplacian operator as a regularizer and the other uses regularization of all components in Fourier space. We compare the performance of all the estimators by means of simulations and noise power analysis. A general feature of these unbiased methods is an enhancement of the low frequency power which, for some of the methods, can be quite severe. We find the best performance is provided by the maximum likelihood method with Fourier space regularization although some of the other methods perform almost as well.
In this review I will describe progress that has been made in determining masses of galaxy clusters using `weak lensing' and how this technique my be applied in the future to determine the dark matter distribution both on supercluster scales and on the scale of galaxy haloes.
Measurement of the gravitational distortion of images of distant galaxies is rapidly becoming established as a powerful probe of the dark mass distribution in clusters of galaxies. With the advent of large mosaics of CCD's these methods should provide a composite total mass profile for galaxy haloes and also measure the power spectrum of mass fluctuations on supercluster scales. We describe how HST observations have been used to place the observational measurement of the shear on a quantitative footing. By artifically stretching and then degrading WFPC2 images to simulate ground based observing it is now possible to directly calibrate the effect of atmospheric seeing. Similar experiments show that one can remove the effect of artificial image anisotropy arising in the atmosphere or telescope. There have also been important advances in the interpretation of the shear: reconstruction techniques have been extended to encompass the strong distortion regime of giant arcs etc., progress has been made in removing a bias present in earlier reconstruction techniques, and we describe new techniques for `aperture densitometry'. We present some new results on clusters of galaxies, and discuss the intimate connections between weak lensing and deep spectroscopy.
We develop and test a method for measuring the gravitational lensing induced distortion of faint background galaxies. We first describe how we locate the galaxies and measure a 2-component `polarisation' or ellipticity statistic $e_α$ whose expectation value should be proportional to the gravitational shear $γ_α$. We then show that an anisotropic instrumental psf perturbs the polarisation by $δe_α= P^s_{αβ} p_β$, where $p_α$ is a measure of the psf anisotropy and $P^s_{αβ}$ is the `linearised smear polarisability tensor'. By estimating $P^s_{αβ}$ for each object we can determine $p_α$ from the foreground stars and apply a correction $-P^s_{αβ}p_β$ to the galaxies. We test this procedure using deep high-resolution images from HST which are smeared with an anisotropic psf and then have noise added to simulate ground-based observations. We find that the procedure works very well. A similar analysis yields a linear shear polarisability tensor $P^γ_{αβ}$ which describes the response to a gravitational shear. This calibrates the polarisation-shear relation, but only for galaxies which are well resolved. To empirically calibrate the effect of seeing on the smaller galaxies we artificially stretch HST images to simulate lensing and then degrade them as before. These experiments provide a rigorous and exacting test of the method under realistic conditions. They show that it is possible to remove the effect of instrumental psf anisotropy, and that the method provides an efficient and quantitative measurement of the gravitational shear.
We develop a method for general non-linear cluster lens reconstruction using the observable distortion of background galaxies. The distortion measures the combination $γ/(1-κ)$ of shear $γ$ and surface density $κ$. From this we obtain an expression for the gradient of $\log (1 - κ)$ in terms of directly measurable quantities. This allows one to reconstruct $1 - κ$ up to an arbitrary constant multiplier. Recent work has emphasised an ambiguity in the relation between the distortion and $γ/(1-κ)$. Here we show that the functional relation depends only on the parity of the images, so if one has data extending to large radii, and if the critical lines can be visually identified (as lines along which the distortion diverges), this ambiguity is resolved. Moreover, we show that for a generic 2-dimensional lens it is possible to locally determine the parity from the distortion. The arbitrary multiplier, which may in fact take a different value in each region bounded by the contour $κ= 1$, can be determined by requiring that the mean surface excess vanish at large radii and that the gradient of $κ$ should be continuous across $κ= 1$. We show how these ideas might be implemented to reconstruct the surface density, if necessary without use of the data in regions where determination of the parity is insecure, and we show how one can measure the mass contained within an aperture, again, if necessary, without using data within the aperture.
Massive clusters of galaxies gravitationally shear the images of faint background galaxies. At large impact parameters the shear is weak, but can still be measured, to a reasonable degree of significance, as a statistical anisotropy of the faint galaxy images. We describe techniques for measuring the shear and discuss the interpretation of the shear field. We have applied this analysis to ms1224+007. We find a clear detection of the shear, but, puzzlingly, we find a mass about three times that obtained by application of the virial theorem, and obtain a very large mass-to-light ratio. Similar results have been obtained by other groups, and we discuss their implications.
We explore the dark matter distribution in ms1224.7+2007 using the gravitational distortion of the images of faint background galaxies. Projected mass image reconstruction reveals a highly significant concentration coincident with the X-ray and optical location. The concentration is seen repeatably in reconstructions from independent subsamples and the azimuthally averaged tangential shear pattern is also clearly seen in the data. The projected mass within a $2.76'$ radius aperture is $\simeq 3.5 \times 10^{14} h^{-1} M_{\sun}$. This is $\simeq 3$ times larger than that predicted if mass traces light with $M/L = 275 h$ as derived from virial analysis. It is very hard to attribute the discrepancy to a statistical fluctuation, and a further indication of a significant difference between the mass and the light comes from a second mass concentration which is again seen in independent subsamples but which is not seen at all in the cluster light. We find a mass per galaxy visible to $I = 22$ of $\simeq 8 \times 10^{12}h^{-1}M_{\sun}$ which, if representative of the universe, implies a density parameter $Ω\sim 2$. We find a null detection of any net shear from large-scale structure with a precision of 0.9\% per component. This is much smaller than the possible detection in a recent comparable study, and the precision here is comparable to to the minimum level of rms shear fluctuations implied by observed large-scale structure.
I review a number of tests of theories for structure formation. Large-scale flows and IRAS galaxies indicate a high density parameter $Ω\simeq 1$, in accord with inflationary predictions, but it is not clear how this meshes with the uniformly low values obtained from virial analysis on scales $\sim$ 1Mpc. Gravitational distortion of faint galaxies behind clusters allows one to construct maps of the mass surface density, and this should shed some light on the large vs small-scale $Ω$ discrepancy. Power spectrum analysis reveals too red a spectrum (compared to standard CDM) on scales $λ\sim 10-100$ $h^{-1}$Mpc, but the gaussian fluctuation hypothesis appears to be in good shape. These results suggest that the problem for CDM lies not in the very early universe --- the inflationary predictions of $Ω= 1$ and gaussianity both seem to be OK; furthermore, the COBE result severely restricts modifications such as tilting the primordial spectrum --- but in the assumed matter content. The power spectrum problem can be solved by invoking a cocktail of mixed dark matter. However, if gravitational lensing fails to reveal extended dark mass around clusters then we may be forced to explore more radical possibilities for the dark matter.
We develop a general method for power spectrum analysis of three dimensional redshift surveys. We present rigorous analytical estimates for the statistical uncertainty in the power and we are able to derive a rigorous optimal weighting scheme under the reasonable (and largely empirically verified) assumption that the long wavelength Fourier components are Gaussian distributed. We apply the formalism to the updated 1-in-6 QDOT IRAS redshift survey, and compare our results to data from other probes: APM angular correlations; the CfA and the Berkeley 1.2Jy IRAS redshift surveys. Our results bear out and further quantify the impression from e.g.\ counts-in-cells analysis that there is extra power on large scales as compared to the standard CDM model with $Ωh\simeq 0.5$. We apply likelihood analysis using the CDM spectrum with $Ωh$ as a free parameter as a phenomenological family of models; we find the best fitting parameters in redshift space and transform the results to real space. Finally, we calculate the distribution of the estimated long wavelength power. This agrees remarkably well with the exponential distribution expected for Gaussian fluctuations, even out to powers of ten times the mean. Our results thus reveal no trace of periodicity or other non-Gaussian behavior.