The Adaptive TreePM: an adaptive resolution code for cosmological N-body simulations

Cosmological N -body simulations are used for a variety of applications. Indeed progress in the study of large-scale structures and galaxy formation would have been very limited without this tool. For nearly 20 yr the limitations imposed by computing power forced simulators to ignore some of the basic requirements for modelling gravitational instability. One of the limitations of most cosmological codes has been the use of a force softening length that is much smaller than the typical interparticle separation. This leads to departures from collisionless evolution that is desired in these simulations. We propose a particle-based method with an adaptive resolution where the force softening length is reduced in high-density regions while ensuring that it remains well above the local interparticle separation. The method, called the Adaptive TreePM (ATreePM), is based on the TreePM code. We present the mathematical model and an implementation of this code, and demonstrate that the results converge over a range of options for parameters introduced in generalizing the code from the TreePM code. We explicitly demonstrate collisionless evolution in collapse of an oblique plane wave. We compare the code with the fixed resolution TreePM code and also an implementation that mimics adaptive mesh refinement methods and comment on the agreement and disagreements in the results. We find that in most respects the ATreePM code performs at least as well as the fixed resolution TreePM in highly overdense regions, from clustering and number density of haloes to internal dynamics of haloes. We also show that the adaptive code is faster than the corresponding high-resolution TreePM code.     

The three-point function in large-scale structure – I. The weakly non-linear regime in N-body simulations

This paper presents a comparison of the predictions for the two- and three-point correlation functions of density fluctuations, ξ and ζ , in gravitational perturbation theory (PT) against large cold dark matter (CDM) simulations. This comparison is made possible for the first time on large weakly non-linear scales (>10  h ⁻¹ Mpc) thanks to the development of a new algorithm for estimating correlation functions for millions of points in only a few minutes. Previous studies in the literature comparing the PT predictions of the three-point statistics with simulations have focused mostly on Fourier space, angular space or smoothed fields. Results in configuration space, such as those presented here, were limited to small scales where leading-order PT gives a poor approximation. Here we also propose and apply a method for separating the first-order and subsequent contributions to PT by combining different output times from the evolved simulations. We find that in all cases there is a regime where simulations do reproduce the leading-order (tree-level) predictions of PT for the reduced three-point function   Q ₃∼ ζ / ξ ²  . For steeply decreasing correlations (such as the standard CDM model) deviations from the tree-level results are important even at relatively large scales, ≃20 Mpc  h ⁻¹. On larger scales ξ goes to zero and the results are dominated by sampling errors. In more realistic models (such as the ΛCDM cosmology) deviations from the leading-order PT become important at smaller scales   r ≃10 Mpc  h ^-1  , although this depends on the particular three-point configuration. We characterize the range of validity of this agreement and show the behaviour of the next-order (one-loop) corrections.     

初始谱指数为-1的非线性重谱(Bispectrum)

重谱即傅立叶空间的三点相关函数，是研究由引力不稳定性产生的大尺度结构的最低非高斯阶统计性质．在傅立叶空间中，发现二维和三维空间的统计性质是等价的，相对三维快速傅立叶，二维快速傅立叶变换可以节省很多计算机资源．结合二维和三维的快速傅立叶变换，利用高精度的初始谱指数为-1的N体数值模拟，测量了从弱线性区域到强非线性区域的重谱．在测量过程中，考虑并严格校正了作用于功率谱和重谱上的数值效应，如离散效应（particle discreteness）、软化因子（force softening）、模拟盒子的大小、混淆效应（alias effect）．发现约化重谱即使在强非线性区域仍然依赖于三角形的形状和尺度，并随着波数的增加而增加，这和前人的假设完全不同．在测量结果的基础上，拟合了重谱的一个适用于初始谱指数为-1的宇宙学模型的经验公式．     

Accurate estimators of power spectra in N-body simulations

A method to rapidly estimate the Fourier power spectrum of a point distribution is presented. This method relies on a Taylor expansion of the trigonometric functions. It yields the Fourier modes from a number of fast Fourier transforms (FFTs), which is controlled by the order N of the expansion and by the dimension D of the system. In three dimensions, for the practical value   N = 3  , the number of FFTs required is 20.
We apply the method to the measurement of the power spectrum of a periodic point distribution that is a local Poisson realization of an underlying stationary field. We derive an explicit analytic expression for the spectrum, which allows us to quantify – and correct for – the biases induced by discreteness and by the truncation of the Taylor expansion, and to bound the unknown effects of aliasing of the power spectrum. We show that these aliasing effects decrease rapidly with the order N . For   N = 3  , they are expected to be, respectively, smaller than  ∼10⁻⁴  and 0.02 at half the Nyquist frequency and at the Nyquist frequency of the grid used to perform the FFTs. The only remaining significant source of errors is reduced to the unavoidable cosmic/sample variance due to the finite size of the sample.
The analytical calculations are successfully checked against a cosmological N -body experiment. We also consider the initial conditions of this simulation, which correspond to a perturbed grid. This allows us to test a case where the local Poisson assumption is incorrect. Even in that extreme situation, the third-order Fourier–Taylor estimator behaves well, with aliasing effects restrained to at most the per cent level at half the Nyquist frequency.
We also show how to reach arbitrarily large dynamic range in Fourier space (i.e. high wavenumber), while keeping statistical errors in control, by appropriately 'folding' the particle distribution.     

TreePM: A code for cosmological N-body simulations

We describe the TreePM method for carrying out large N-Body simulations to study formation and evolution of the large scale structure in the Universe. This method is a combination of Barnes and Hut tree code and Particle-Mesh code. It combines the automatic inclusion of periodic boundary conditions of PM simulations with the high resolution of tree codes. This is done by splitting the gravitational force into a short range and a long range component. We describe the splitting of force between these two parts. We outline the key differences between TreePM and some other N-Body methods.     

Multimass spherical structure models for N-body simulations

We present a simple and efficient method to set up spherical structure models for N -body simulations with a multimass technique. This technique reduces by a substantial factor the computer run time needed in order to resolve a given scale as compared to single-mass models. It therefore allows to resolve smaller scales in N -body simulations for a given computer run time. Here, we present several models with an effective resolution of up to  1.68 × 10⁹  particles within their virial radius which are stable over cosmologically relevant time-scales. As an application, we confirm the theoretical prediction by Dehnen that in mergers of collisionless structures like dark matter haloes always the cusp of the steepest progenitor is preserved. We model each merger progenitor with an effective number of particles of approximately 10⁸ particles. We also find that in a core–core merger the central density approximately doubles whereas in the cusp–cusp case the central density only increases by approximately 50 per cent. This may suggest that the central regions of flat structures are better protected and get less energy input through the merger process.     

N-body simulations of the Magellanic stream

A suite of high-resolution N -body simulations of the Magellanic Clouds–Milky Way system are presented and compared directly with newly available data from the H  i Parkes All-Sky Survey (HIPASS). We show that the interaction between Small Magellanic Clouds (SMC) and Large Magellanic Clouds results in both a spatial and kinematical bifurcation of both the stream and the leading arm. The spatial bifurcation of the stream is readily apparent in the HIPASS data, and the kinematical bifurcation is also tentatively identified. This bifurcation provides strong support for the tidal disruption origin for the Magellanic stream. A fiducial model for the Magellanic Clouds (MCs) is presented upon completion of an extensive parameter survey of the potential orbital configurations of the MCs and the viable initial boundary conditions for the disc of the SMC. The impact of the choice of these critical parameters upon the final configurations of the stream and leading arm is detailed.     

Post-Newtonian N-body simulations

We report on the first fully consistent conventional cluster simulation which includes terms up to the third-order post-Newtonian approximation. Numerical problems for treating extremely energetic binaries orbiting a single massive object are circumvented by employing the special 'wheel-spoke' regularization method of Zare which has not been used in large- N simulations before. Idealized models containing   N = 1 × 10⁵  particles of mass  1 M_⊙  with a central black hole (BH) of  300 M_⊙  have been studied on GRAPE-type computers. An initial half-mass radius of   r _h≃ 0.1  pc is sufficiently small to yield examples of relativistic coalescence. This is achieved by significant binary shrinkage within a density cusp environment, followed by the generation of extremely high eccentricities which are induced by Kozai cycles and/or resonant relaxation. More realistic models with white dwarfs and 10 times larger half-mass radii also show evidence of general relativity effects before disruption. An experimentation with the post-Newtonian terms suggests that reducing the time-scales for activating the different orders progressively may be justified for obtaining qualitatively correct solutions without aiming for precise predictions of the final gravitational radiation wave form. The results obtained suggest that the standard loss-cone arguments underestimate the swallowing rate in globular clusters containing a central BH.