On the least action principle in cosmology

Given the present distribution of mass tracing objects in an expanding universe, we develop and test a fast method for recovering their past orbits using the least action principle. In this method, termed FAM for fast action minimization, the orbits are expanded in a set of orthogonal time basis functions satisfying the appropriate boundary conditions at the initial and final times. The conjugate gradient method is applied to locate the extremum of the action in the space of the expansion coefficients of the orbits. The treecode gravity solver routine is used for computing the gravitational field appearing in the action and the potential field appearing in the gradient of the action. The time integration of the Lagrangian is done using Gaussian quadratures. FAM allows us to increase the number of galaxies over previous numerical action principle implementations by more than one order of magnitude. For example, orbits for the 15 000 IRAS PSC z galaxies can be recovered in 12 000 CPU seconds on a 400-MHz DEC-Alpha machine. FAM can recover the present peculiar velocities of particles and the initial fluctuations field. It successfully recovers the flow field down to cluster scales, where deviations of the flow from the Zel'dovich solution are significant. We also show how to recover orbits from the present distribution of objects in redshift space by direct minimization of a modified action, without iterating the solution between real and redshift spaces.     

Simulations of deep pencil-beam redshift surveys

We create mock pencil-beam redshift surveys from very large cosmological N -body simulations of two cold dark matter (CDM) cosmogonies, an Einstein–de Sitter model ( τ CDM) and a flat model with Ω₀=0.3 and a cosmological constant (ΛCDM). We use these to assess the significance of the apparent periodicity discovered by Broadhurst et al. Simulation particles are tagged as 'galaxies' so as to reproduce observed present-day correlations. They are then identified along the past light-cones of hypothetical observers to create mock catalogues with the geometry and the distance distribution of the Broadhurst et al. data. We produce 1936 (2625) quasi-independent catalogues from our τ CDM (ΛCDM) simulation. A couple of large clumps in a catalogue can produce a high peak at low wavenumbers in the corresponding one-dimensional power spectrum, without any apparent large-scale periodicity in the original redshift histogram. Although the simulated redshift histograms frequently display regularly spaced clumps, the spacing of these clumps varies between catalogues and there is no 'preferred' period over our many realizations. We find only a 0.72 (0.49) per cent chance that the highest peak in the power spectrum of a τ CDM (ΛCDM) catalogue has a peak-to-noise ratio higher than that in the Broadhurst et al. data. None of the simulated catalogues with such high peaks shows coherently spaced clumps with a significance as high as that of the real data. We conclude that in CDM universes, the regularity on a scale of ∼130  h ⁻¹ Mpc observed by Broadhurst et al. has a priori probability well below 10⁻³.     

Gravitational collapse in an expanding background and the role of substructure – II. Excess power at small scales and its effect on collapse of structures at larger scales

We study the interplay of clumping at small scales with the collapse and relaxation of perturbations at larger scales using N -body simulations. We quantify the effect of collapsed haloes on perturbations at larger scales using the two-point correlation function, moments of counts in cells and the mass function. The purpose of the study is twofold and the primary aim is to quantify the role played by collapsed low-mass haloes in the evolution of perturbations at large scales; this is in view of the strong effect seen when the large scale perturbation is highly symmetric. Another reason for this study is to ask whether features or a cut-off in the initial power spectrum can be detected using measures of clustering at scales that are already non-linear. The final aim is to understand the effect of ignoring perturbations at scales smaller than the resolution of N -body simulations. We find that these effects are ignorable if the scale of non-linearity is larger than the average interparticle separation in simulations. Features in the initial power spectrum can be detected easily if the scale of these features is in the linear regime; detecting such features becomes difficult as the relevant scales become non-linear. We find no effect of features in initial power spectra at small scales on the evolved power spectra at large scales. We may conclude that, in general, the effect on the evolution of perturbations at large scales of clumping on small scales is very small and may be ignored in most situations.     

Constraining cold dark matter halo merger rates using the coagulation equations

We place additional constraints on the three parameters of the dark matter halo merger rate function recently proposed by Parkinson, Cole & Helly by utilizing Smoluchowski's coagulation equation, which must be obeyed by any binary merging process which conserves mass. We find that the constraints from Smoluchowski's equation are degenerate, limiting to a thin plane in the three-dimensional parameter space. This constraint is consistent with those obtained from fitting to N -body measures of progenitor mass functions, and provides a better match to the evolution of the overall dark matter halo mass function, particularly for the most massive haloes. We demonstrate that the proposed merger rate function does not permit an exact solution of Smoluchowski's equation and, therefore, the choice of parameters must reflect a compromise between fitting various parts of the mass function. The techniques described herein are applicable to more general merger rate functions, which may permit a more accurate solution of Smoluchowski's equation. The current merger rate solutions are most probably sufficiently accurate for the vast majority of applications.     

Gravitational collapse in an expanding background and the role of substructure – I. Planar collapse

We study the interplay of clumping at small scales with the collapse and relaxation of perturbations at much larger scales. We present results of our analysis when the large-scale perturbation is modelled as a plane wave. We find that in the absence of substructure, collapse leads to formation of a pancake with multistream regions. Dynamical relaxation of the plane wave is faster in the presence of substructure. Scattering of substructures and the resulting enhancement of transverse motions of haloes in the multistream region lead to a thinner pancake. In turn, collapse of the plane wave leads to formation of more massive collapsed haloes as compared to the collapse of substructure in the absence of the plane wave. The formation of more massive haloes happens without any increase in the total mass in collapsed haloes. A comparison with the Burgers equation approach in the absence of any substructure suggests that the preferred value of effective viscosity depends primarily on the number of streams in a region.     

The 3D skeleton: tracing the filamentary structure of the Universe

The skeleton formalism, which aims at extracting and quantifying the filamentary structure of our Universe, is generalized to 3D density fields. A numerical method for computing a local approximation of the skeleton is presented and validated here on Gaussian random fields. It involves solving equation     , where  ∇ρ  and     are the gradient and Hessian matrix of the field. This method traces well the filamentary structure in 3D fields such as those produced by numerical simulations of the dark matter distribution on large scales, and is insensitive to monotonic biasing.
Two of its characteristics, namely its length and differential length, are analysed for Gaussian random fields. Its differential length per unit normalized density contrast scales like the probability distribution function of the underlying density contrast times the total length times a quadratic Edgeworth correction involving the square of the spectral parameter. The total length-scales like the inverse square smoothing length, with a scaling factor given by  0.21 (5.28 + n )  where n is the power index of the underlying field. This dependency implies that the total length can be used to constrain the shape of the underlying power spectrum, hence the cosmology.
Possible applications of the skeleton to galaxy formation and cosmology are discussed. As an illustration, the orientation of the spin of dark haloes and the orientation of the flow near the skeleton is computed for cosmological dark matter simulations. The flow is laminar along the filaments, while spins of dark haloes within 500 kpc of the skeleton are preferentially orthogonal to the direction of the flow at a level of 25 per cent.     

Non-linear redshift distortions: the two-point correlation function

We consider a situation where the density and peculiar velocities in real space are linear, and we calculate ξ _s , the two-point correlation function in redshift space, incorporating all non-linear effects which arise as a consequence of the map from real to redshift space. Our result is non-perturbative and it includes the effects of possible multi-streaming in redshift space. We find that the deviations from the predictions of the linear redshift distortion analysis increase for the higher spherical harmonics of ξ _s . While the deviations are insignificant for the monopole ξ ₀, the hexadecapole ξ ₄ exhibits large deviations from the linear predictions. For a COBE normalized     ,     cold dark matter (CDM) power spectrum, our results for ξ ₄ deviate from the linear predictions by a factor of two on the scale of ∼10  h ⁻¹ Mpc. The deviations from the linear predictions depend separately on f (Ω) and b . This holds the possibility of removing the degeneracy that exists between these two parameters in the linear analysis of redshift surveys which yields only     .
We also show that the commonly used phenomenological model, where the non-linear redshift two-point correlation function is calculated by convolving the linear redshift correlation function with an isotropic pair velocity distribution function, is a limiting case of our result.     

Self-similar spherical collapse with non-radial motions

We derive the asymptotic mass profile near the collapse centre of an initial spherical density perturbation, δ ∝ M ^{− ε} , of collisionless particles with non-radial motions. We show that angular momenta introduced at the initial time do not affect the mass profile. Alternatively, we consider a scheme in which a particle moves on a radial orbit until it reaches its turnaround radius, r ∗. At turnaround the particle acquires an angular momentum L =ℒ√ GM _* r _* per unit mass, where M ∗ is the mass interior to r ∗. In this scheme, the mass profile is M ∝ r ^{3/(1+3 ε )} for all ε >0 , in the region r / r _t ≪ℒ , where r _t is the current turnaround radius. If ℒ≪1 then the profile in the region ℒ≪ r / r _t ≪1 is M ∝ r for ε <2/3 , and remains M ∝ r ^{3/(1+3 ε )} for ε ≥2/3 . The derivation relies on a general property of non-radial orbits which is that the ratio of the pericentre to apocentre is constant in a force field k ( t ) r ⁿ with k ( t ) varying adiabatically.