Cosmological perturbation theory and the spherical collapse model — III. The velocity divergence field and the Ω dependence

Cosmological perturbation theory (PT) is a useful tool to study the cumulants of the density and velocity fields in the large-scale structure of the Universe. In Papers I and II of this series we saw that the spherical collapse (SC) model provides the exact solution to PT at tree-level and gives a good approximation to the loop corrections (next-to-leading orders), indicating negligible tidal effects. Here, we derive predictions for the (smoothed) cumulants of the velocity divergence field θ ≡ ▽ ⊙  v for an irrotational fluid in the SC model. By comparing these with the exact analytic results of Scoccimarro &38; Frieman, it is shown that, at least for the unsmoothed case, the loop corrections to the cumulants of θ are dominated by tidal effects. However, most of the tidal contribution seems to cancel out when computing the hierarchical ratios, T _J  = 〈θ ^J 〉 / 〈θ²〉  ^{J −1}. We also extend the work presented in Papers I and II to give predictions for the cumulants of the density and velocity divergence fields in non-flat spaces. In particular, we show the equivalence between the spherically symmetric solution to the equations of motion in the SC model (given in terms of the density) and that of the Lagrangian PT approach (given in terms of the displacement field). It is shown that the Ω dependence is very weak for both cosmic fields even at one loop (a 10 per cent effect at most), except for the overall factor f (Ω) that couples to the velocity divergence.     

The non-linear redshift-space power spectrum of galaxies

We study the power spectrum of galaxies in redshift space, with third-order perturbation theory to include corrections that are absent in linear theory. We assume a local bias for the galaxies: i.e., the galaxy density is sampled from some local function of the underlying mass distribution. We find that the effect of the non-linear bias in real space is to introduce two new features: first, there is a contribution to the power which is constant with wavenumber, whose nature we reveal as essentially a shot-noise term. In principle this contribution can mask the primordial power spectrum, and could limit the accuracy with which the latter might be measured on very large scales. Secondly, the effect of second- and third-order bias is to modify the effective bias (defined as the square root of the ratio of galaxy power spectrum to matter power spectrum). The effective bias is almost scale-independent over a wide range of scales. These general conclusions also hold in redshift space. In addition, we have investigated the distortion of the power spectrum by peculiar velocities, which may be used to constrain the density of the Universe. We look at the quadrupole-to-monopole ratio, and find that higher order terms can mimic linear theory bias, but the bias implied is neither the linear bias, nor the effective bias referred to above. We test the theory with biased N -body simulations, and find excellent agreement in both real and redshift space, providing the local biasing is applied on a scale whose fractional rms density fluctuations are < 0.5.     

Spatial dynamics of retail structure and the venerable retail hierarchy

The retail system is continuously adapting itself to changing demographic conditions, consumer behavior and economic conditions. Existing retail units, getting larger in order to achieve economics of scale, and new forms of large-scale retailing do not fit easily in the traditional spatial pattern of retail concentrations. Neverteless the spatial distribution of retailing in urban areas, as schematized by Berry thirty-five years ago, still has its value. Through an upward hierarchical movement the retail system has adapted itself to an inflexible spatial system, in which downtown has still a dominant position. Future developments, however, may weaken the position of central shopping areas in favour of peripheral locations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Testing deprojection algorithms on mock angular catalogues: evidence for a break in the power spectrum

We produce mock angular catalogues from simulations with different initial power spectra to test methods that recover measures of clustering in three dimensions, such as the power spectrum, variance and higher order cumulants. We find that the statistical properties derived from the angular mock catalogues are in good agreement with the intrinsic clustering in the simulations. In particular, we concentrate on the detailed predictions for the shape of the power spectrum, P ( k ). We find that there is good evidence for a break in the galaxy P ( k ) at scales in the range 0.02< k <0.06 h Mpc⁻¹, using an inversion technique applied to the angular correlation function measured from the APM Galaxy Survey. For variants on the standard cold dark matter (CDM) model, a fit at the location of the break implies Ω h =0.45±0.10, where Ω is the ratio of the total matter density to the critical density, and Hubble's constant is parametrized as H ₀=100 h km s⁻¹ Mpc⁻¹. On slightly smaller, though still quasi-linear scales, there is a feature in the APM power spectrum where the local slope changes appreciably, with the best match to CDM models obtained for Ω h ≃0.2. Hence the location and narrowness of the break in the APM power spectrum combined with the rapid change in its slope on quasi-linear scales cannot be matched by any variant of CDM, including models that have a non-zero cosmological constant or a tilt to the slope of the primordial P ( k ). These results are independent of the overall normalization of the CDM models or any simple bias that exists betwen the galaxy and mass distributions.     

Modelling the X-ray cluster dipole and cluster contribution to the soft X-ray background

We investigate the sampling and dipole convergence properties of flux-limited samples of mock X-ray clusters in relation to their underlying 'parent' cluster distribution. To this purpose, we resort to numerical simulations of the cluster distribution and extract samples resembling the main observational features of X-ray selected cluster samples. The flux-limited samples, being quite sparse, underestimate the amplitude of the 'parent' cluster dipole by ≈ 15 per cent on average for Local Group-like observers. However, the general shapes of their dipole amplitude profiles are in relatively good agreement. We also calculate the expected contribution of clusters, selected according to the relevant criteria, to the soft (i.e. 0.1–2.4 keV) extragalactic X-ray background (XRB), using the ESO Key Project X-ray luminosity function, assuming a flat universe with vanishing cosmological constant. We obtain a value of about 10 per cent of the observed XRB flux.     

Submillimetre-wave gravitational lenses and cosmology

One of the most direct routes for investigating the geometry of the Universe is provided by the numbers of strongly magnified gravitationally lensed galaxies as compared with those that are either weakly magnified or de-magnified. In the submillimetre waveband the relative abundance of strongly lensed galaxies is expected to be larger as compared with the optical or radio wavebands, both in the field and in clusters of galaxies. The predicted numbers depend on the properties of the population of faint galaxies in the submillimetre waveband, which was formerly very uncertain; however, recent observations of lensing clusters have reduced this uncertainty significantly and confirm that a large sample of galaxy–galaxy lenses could be detected and investigated using forthcoming facilities, including the FIRST and Planck Surveyor space missions and a large ground-based millimetre/submillimetre-wave interferometer array (MIA). We discuss how this sample could be used to impose limits on the values of cosmological parameters and the total density and form of evolution of the mass distribution of bound structures, even in the absence of detailed lens modelling for individual members of the sample. The effects of different world models on the form of the magnification bias expected in sensitive submillimetre-wave observations of clusters are also discussed, because an MIA could resolve and investigate images in clusters in detail.