Measuring the three-dimensional shear from simulation data, with applications to weak gravitational lensing

We have developed a new three-dimensional algorithm, based on the standard P³M method, for computing deflections resulting from weak gravitational lensing. We compare the results of this method with those of the two-dimensional planar approach, and rigorously outline the conditions under which the two approaches are equivalent. Our new algorithm uses a Fast Fourier Transform convolution method for speed, and has a variable softening feature to provide a realistic interpretation of the large-scale structure in a simulation. The output values of the code are compared with those from the Ewald summation method, which we describe and develop in detail. With an optimal choice of the high-frequency filtering in the Fourier convolution, the maximum errors, when using only a single particle, are about 7 per cent, with an rms error less than 2 per cent. For ensembles of particles, used in typical N -body simulations, the rms errors are typically 0.3 per cent. We describe how the output from the algorithm can be used to generate distributions of magnification, source ellipticity, shear and convergence for large-scale structure.     

Reconstruction of cosmological initial conditions from galaxy redshift catalogues

We present and test a new method for the reconstruction of cosmological initial conditions from a full-sky galaxy catalogue. This method, called ZTRACE, is based on a self-consistent solution of the growing mode of gravitational instabilities according to the Zel'dovich approximation and higher order in Lagrangian perturbation theory. Given the evolved redshift-space density field, smoothed on some scale, ZTRACE finds, via an iterative procedure, an approximation to the initial density field for any given set of cosmological parameters; real-space densities and peculiar velocities are also reconstructed. The method is tested by applying it to N -body simulations of an Einstein–de Sitter and an open cold dark matter universe. It is shown that errors in the estimate of the density contrast dominate the noise of the reconstruction. As a consequence, the reconstruction of real-space density and peculiar velocity fields using non-linear algorithms is little improved over those based on linear theory. The use of a mass-preserving adaptive smoothing, equivalent to a smoothing in Lagrangian space, allows an unbiased (although noisy) reconstruction of initial conditions, as long as the (linearly extrapolated) density contrast does not exceed unity. The probability distribution function of the initial conditions is recovered to high precision, even for Gaussian smoothing scales of ∼5  h ⁻¹ Mpc, except for the tail at δ ≥1. This result is insensitive to the assumptions of the background cosmology.     

The Gaussian cell two-point 'energy-like' equation: application to large-scale galaxy redshift and peculiar motion surveys

We introduce a simple linear equation relating the line-of-sight peculiar-velocity and density contrast correlation functions. The relation, which we call the Gaussian cell two-point 'energy-like' equation , is valid at the distant-observer limit and requires Gaussian smoothed fields. In the variance case, i.e. at zero lag, the equation is similar in its mathematical form to the Irvine–Layzer cosmic energy equation. β estimation with this equation from the Point Source Catalogue Redshift (PSC) survey and the SEcat catalogue of peculiar velocities is carried out, returning a value of  β= 0.44 ± 0.08  . The applicability of the method for the 6dF galaxy redshift and peculiar motions survey is demonstrated with mock data where it is shown that β could be determined with ≈10 per cent accuracy. The prospects for constraining the dark energy equation of state with this method from the kinematic and thermal Sunyaev–Zel'dovich cluster surveys are discussed. The equation is also used to construct a non-parametric mass-density power-spectrum estimator from peculiar-velocity data.     

Power spectrum analysis of the ESO Slice Project galaxy redshift survey

We measure the power spectrum of the galaxy distribution in the ESO Slice Project (ESP) galaxy redshift survey. We develop a technique to describe the survey window function analytically, and then deconvolve it from the measured power spectrum using a variant of the Lucy method. We test the whole deconvolution procedure on ESP mock catalogues drawn from large N -body simulations, and find that it is reliable for recovering the correct amplitude and shape of P ( k ) at k >0.065  h  Mpc⁻¹. In general, the technique is applicable to any survey composed of a collection of circular fields with an arbitrary pattern on the sky, as typical of surveys based on fibre spectrographs. The estimated power spectrum has a well-defined power-law shape k ⁿ with n ≃−2.2 for k ≥0.2  h  Mpc⁻¹, and a smooth bend to a flatter shape ( n ≃−1.6) for smaller k . The smallest wavenumber where a meaningful reconstruction can be performed ( k ∼0.06  h  Mpc⁻¹) does not allow us to explore the range of scales where other power spectra seem to show a flattening and hint at a turnover. We also find, by a direct comparison of the Fourier transforms, that the estimate of the two-point correlation function ξ ( s ) is much less sensitive to the effect of a problematic window function, such as that of the ESP, than the power spectrum. Comparison with other surveys shows an excellent agreement with estimates from blue-selected surveys. In particular, the ESP power spectrum is virtually indistinguishable from that of the Durham–UKST survey over the common range of k , an indirect confirmation of the quality of the deconvolution technique applied.     

Non-Gaussian cosmic microwave background temperature fluctuations from peculiar velocities of clusters

We use numerical simulations of a (480 Mpc  h ⁻¹)³ volume to show that the distribution of peak heights in maps of the temperature fluctuations from the kinematic and thermal Sunyaev–Zeldovich (SZ) effects will be highly non-Gaussian, and very different from the peak-height distribution of a Gaussian random field. We then show that it is a good approximation to assume that each peak in either SZ effect is associated with one and only one dark matter halo. This allows us to use our knowledge of the properties of haloes to estimate the peak-height distributions. At fixed optical depth, the distribution of peak heights resulting from the kinematic effect is Gaussian, with a width that is approximately proportional to the optical depth; the non-Gaussianity comes from summing over a range of optical depths. The optical depth is an increasing function of halo mass and the distribution of halo speeds is Gaussian, with a dispersion that is approximately independent of halo mass. This means that observations of the kinematic effect can be used to put constraints on how the abundance of massive clusters evolves, and on the evolution of cluster velocities. The non-Gaussianity of the thermal effect, on the other hand, comes primarily from the fact that, on average, the effect is larger in more massive haloes, and the distribution of halo masses is highly non-Gaussian. We also show that because haloes of the same mass may have a range of density and velocity dispersion profiles, the relation between halo mass and the amplitude of the thermal effect is not deterministic, but has some scatter.     

The probability distribution of cluster formation times and implied Einstein radii

We provide a quantitative assessment of the probability distribution function of the concentration parameter of galaxy clusters. We do so by using the probability distribution function of halo formation times, calculated by means of the excursion set formalism, and a formation redshift-concentration scaling derived from results of N -body simulations. Our results suggest that the observed high concentrations of several clusters are quite unlikely in the standard Λ cold dark matter (ΛCDM) cosmological model, but that due to various inherent uncertainties, the statistical range of the predicted distribution may be significantly wider than commonly acknowledged. In addition, the probability distribution function of the Einstein radius of A1689 is evaluated, confirming that the observed value of  ∼45 ± 5 arcsec  is very improbable in the currently favoured cosmological model. If, however, a variance of ∼20 per cent in the theoretically predicted value of the virial radius is assumed, then the discrepancy is much weaker. The measurement of similarly large Einstein radii in several other clusters would pose a difficulty to the standard model. If so, earlier formation of the large-scale structure would be required, in accord with predictions of some quintessence models. We have indeed verified that in a viable early dark energy model large Einstein radii are predicted in as many as a few tens of high-mass clusters.     

Weak gravitational lensing in the standard cold dark matter model, using an algorithm for three-dimensional shear

We investigate the effects of weak gravitational lensing in the standard cold dark matter cosmology, using an algorithm that evaluates the shear in three dimensions. The algorithm has the advantage of variable softening for the particles, and our method allows the appropriate angular diameter distances to be applied to every evaluation location within each three-dimensional simulation box. We investigate the importance of shear in the distance–redshift relation, and find it to be very small. We also establish clearly defined values for the smoothness parameter in the relation, finding its value to be at least 0.83 at all redshifts in our simulations. From our results, obtained by linking the simulation boxes back to source redshifts of 4, we are able to observe the formation of structure in terms of the computed shear, and also note that the major contributions to the shear come from a very broad range of redshifts. We show the probability distributions for the magnification, source ellipticity and convergence, and also describe the relationships amongst these quantities for a range of source redshifts. We find a broad range of magnifications and ellipticities; for sources at a redshift of 4, 97.5 per cent of all lines of sight show magnifications up to 1.39 and ellipticities up to 0.23. There is clear evidence that the magnification is not linear in the convergence, as might be expected for weak lensing, but contains contributions from higher order terms in both the convergence and the shear. Our results for the one-point distribution functions are generally different from those obtained by other authors using two-dimensional (planar) approaches, and we suggest reasons for the differences. Our magnification distributions for sources at redshifts of 1 and 0.5 are also very different from the results used by other authors to assess the effect on the perceived value of the deceleration parameter, and we briefly address this question.     

Structures in the Great Attractor region

To further our understanding of the Great Attractor (GA), we have undertaken a redshift survey using the 2-degree Field (2dF) instrument on the Anglo-Australian Telescope (AAT). Clusters and filaments in the GA region were targeted with 25 separate pointings resulting in approximately 2600 new redshifts. Targets included poorly studied X-ray clusters from the Clusters in the Zone of Avoidance (CIZA) Catalogue as well as the Cen–Crux and PKS 1343−601 clusters, both of which lie close to the classic GA centre. For nine clusters in the region, we report velocity distributions as well as virial and projected mass estimates. The virial mass of CIZA J1324.7−5736, now identified as a separate structure from the Cen–Crux cluster, is found to be  ∼3 × 10¹⁴ M_⊙  , in good agreement with the X-ray inferred mass. In the PKS 1343−601 field, five redshifts are measured of which four are new. An analysis of redshifts from this survey, in combination with those from the literature, reveals the dominant structure in the GA region to be a large filament, which appears to extend from Abell S0639  ( l = 281°, b =+11°)  to  ( l ∼ 5°, b ∼−50°)  , encompassing the Cen–Crux, CIZA J1324.7−5736, Norma and Pavo II clusters. Behind the Norma cluster at   cz ∼ 15 000 km s⁻¹  , the masses of four rich clusters are calculated. These clusters (Triangulum Australis, Ara, CIZA J1514.6−4558 and CIZA J1410.4−4246) may contribute to a continued large-scale flow beyond the GA. The results of these observations will be incorporated into a subsequent analysis of the GA flow.     

The one-point PDF of the initial conditions of our local Universe from the IRAS PSC redshift catalogue

The algorithm ztrace of Monaco & Efstathiou is applied to the IRAS PSCz catalogue to reconstruct the initial conditions of our local Universe with a resolution down to ~5  h ¹ Mpc. The one-point probability distribution function (PDF) of the reconstructed initial conditions is consistent with the assumptions that: (i) IRAS galaxies trace mass on scales of ~5  h ¹ Mpc and (ii) the statistics of the primordial density fluctuations are Gaussian. We use simulated PSCz catalogues, constructed from N -body simulations with Gaussian initial conditions, to show that local non-linear bias can cause the recovered initial PDF (assuming no bias) to be non-Gaussian. However, for plausible bias models, the distortions of the recovered PDF would be difficult to detect using the volume finely sampled by the PSCz catalogue. So, for Gaussian initial conditions, a range of bias models remain compatible with our PSCz reconstruction results.     

Weak gravitational lensing in different cosmologies, using an algorithm for shear in three dimensions

We present the results of weak gravitational lensing statistics in four different cosmological N -body simulations. The data have been generated using an algorithm for the three-dimensional shear, which makes use of a variable softening facility for the N -body particle masses, and enables a physical interpretation for the large-scale structure to be made. Working in three dimensions also allows the correct use of the appropriate angular diameter distances.
Our results are presented on the basis of the filled-beam approximation in view of the variable particle softening scheme in our algorithm. The importance of the smoothness of matter in the Universe for the weak lensing results is discussed in some detail.
The low-density cosmology with a cosmological constant appears to give the broadest distributions for all the statistics computed for sources at high redshifts. In particular, the range in magnification values for this cosmology has implications for the determination of the cosmological parameters from high-redshift type Ia supernovae. The possibility of determining the density parameter from the non-Gaussianity in the probability distribution for the convergence is discussed.