Fast algorithms for potential evaluation in N-body problems often tend to be extremely abstract and complex. This thesis presents a simple, hierarchical approach to solving the potential evaluation problem in O(n) time. The approach is developed in the field of electrostatics and can be extended to N-body problems in general. Herein, the potential vector is expressed as a product of the potential matrix and the charge vector. The potential matrix itself is a product of component matrices. The potential function satisfies the Laplace equation and is hence expressed as a linear combination of spherical harmonics, which form the general solutions of the Laplace equation. The orthogonality of the spherical harmonics is exploited to reduce execution time. The duality of the various lists in the algorithm is used to reduce storage and computational complexity. A smart tree-construction strategy leads to efficient parallelism at computation intensive stages of the algorithm. The computational complexity of the algorithm is better than that of the Fast Multipole Algorithm, which is one of the fastest contemporary algorithms to solve the potential evaluation problem. Experimental results show that accuracy of the algorithm is comparable to that of the Fast Multipole Algorithm. However, this approach uses some implementation principles from the Fast Multipole Algorithm. Parallel efficiency and scalability of the algorithms are studied by the experiments on IBM p690 multiprocessors.

This thesis develops methods to identify periodic solutions to the n-body problem by representing gravitational orbits with Fourier series. To find periodic orbits, a minimization function was developed that compares the second derivative of the Fourier series with Newtonian gravitation acceleration and modifies the Fourier coefficients until the orbits match. Software was developed to minimize the function and identify the orbits using gradient descent and quadratic curves. A Newtonian gravitational simulator was developed to read the initial orbit data and numerically simulate the orbits with accurate motion integration, allowing for comparison to the Fourier series orbits and investigation of their stability. The orbits found with the programs correlate with orbits from literature, and a number remain stable when simulated. / February 2016

We present a precision study of large-scale structure from large galaxy redshift surveys. We focus on two main subjects of large-scale structure: precisioncosmology with baryon acoustic oscillations from large galaxy surveys and the evolution of galaxy clustering for passively flowing galaxies.The baryon acoustic oscillations in galaxy redshift surveys can serve as an efficient standard ruler to measure the cosmological distance scale, i.e., theangular diameter distances and Hubble parameters, as a function of redshift, and therefore dark energy parameters. We use a Fisher matrix formalism to show that such a standard ruler tests can constrain the angular diameter distances and Hubble parameters to a precision of a few percent, thereby providing robust measurements of present-day dark energy density and its time-dependence.We use N-body simulations to investigate possible systematic errors in the recovery of the cosmological distance scale from galaxy redshift surveys. We show that the baryon signature on linear and quasi-linear scales is robust against nonlinear growth, redshift distortions, and halo (or galaxy) bias, albeit partial obscuration of the signature occurs due to nonlinear growth and redshift distortions.We present the improved Fisher matrix formalism which incorporates the Lagrangian displacement field to describe the nonlinear effects on baryon signature as a function of time and scale. We present a physically motivated, reduced 2-dimensional fitting formula for the full Fisher matrix formalism. We show that distance precision from the revised formalism is in excellent agreement with distance precision from N-body simulations.Finally, we present a numerical study of the evolution of galaxy clustering when galaxies flow passively from high redshift to low redshift, that is, without merging or new formations. We show that passive flow evolution induces interesting characteristics in the galaxy distribution at low redshift: we find an asymptotic convergence in galaxy clustering and halo occupation distribution regardless of the initial distribution of galaxies.

Fast algorithms for potential evaluation in N-body problems often tend to be extremely abstract and complex. This thesis presents a simple, hierarchical approach to solving the potential evaluation problem in O(n) time. The approach is developed in the field of electrostatics and can be extended to N-body problems in general. Herein, the potential vector is expressed as a product of the potential matrix and the charge vector. The potential matrix itself is a product of component matrices. The potential function satisfies the Laplace equation and is hence expressed as a linear combination of spherical harmonics, which form the general solutions of the Laplace equation. The orthogonality of the spherical harmonics is exploited to reduce execution time. The duality of the various lists in the algorithm is used to reduce storage and computational complexity. A smart tree-construction strategy leads to efficient parallelism at computation intensive stages of the algorithm. The computational complexity of the algorithm is better than that of the Fast Multipole Algorithm, which is one of the fastest contemporary algorithms to solve the potential evaluation problem. Experimental results show that accuracy of the algorithm is comparable to that of the Fast Multipole Algorithm. However, this approach uses some implementation principles from the Fast Multipole Algorithm. Parallel efficiency and scalability of the algorithms are studied by the experiments on IBM p690 multiprocessors.

We consider the N-body problem given by quasihomogeneous force functions of the form (C_1)/r^a + (C_2)/r^b (C_1, C_2, a, b constants and a, b positive with a less than or equal to b) and address the fundamentals of homographic solutions. Generalizing techniques of the classical N-body problem, we prove necessary and sufficient conditions for a homographic solution to be either homothetic, or relative equilibrium. We further prove an analogue of the Lagrange-Pizzetti theorem based on our techniques. We also study the central configurations for quasihomogeneous force functions and settle the classification and properties of simultaneous and extraneous central configurations. In the last part of the thesis, we combine these findings with the Lagrange-Pizzetti theorem to show the link between homographic solutions and central configurations, to prove the existence of homographic solutions and to give algorithms for their construction. / Graduate

The method of baryon acoustic oscillation (BAO) is among the best probes of the dark energy equation of state, and worldwide efforts are being invested in order to perform measurements that are accurate at the percent level. In current data analyses, however, estimates of the error about the BAO are based on the assumption that the density field can be treated as Gaussian, an assumption that becomes less accurate as smaller scales are included in the measurement. It was recently shown from large samples of N-body simulations that the error bars about the BAO obtained this way are in fact up to 15-20 per cent too small. This important bias has shaken the confidence in the way error bars are calculated, and is motivating developments of analyses pipelines that include non-Gaussian features in the matter density fields. In this thesis, we propose general strategies to incorporate non-Gaussian effects in the context of a survey. After describing the high performance N-body code that we used, we present novel properties of the non-Gaussian uncertainty about the matter power spectrum, and explain how these combine with a general survey selection function. Assuming that the non-Gaussian features that are observed in the simulations correspond to those of Nature, this approach is the first unbiased measurement of the error bar about the power spectrum, which simultaneously removes the undesired bias on the BAO error. We then relax this assumption about the similitude of the non-Gaussian natures in simulations and data, and develop tools that aim at measuring the non-Gaussian error bars exclusively from the data. It is possible to improve the constraining power of non-Gaussian analyses with `Gaussianizations' techniques, which map the observed fields into something more Gaussian. We show that two of such techniques maximally recover degrees of freedom that were lost in the gravitational collapse. Finally, from a large sample of high resolution N-body realizations, we construct a series of weak gravitational lensing distortion maps and provide high resolution halo catalogues that are used by the CFTHLenS community to calibrate their estimators and study many secondary effects with unprecedented accuracy.

The method of baryon acoustic oscillation (BAO) is among the best probes of the dark energy equation of state, and worldwide efforts are being invested in order to perform measurements that are accurate at the percent level. In current data analyses, however, estimates of the error about the BAO are based on the assumption that the density field can be treated as Gaussian, an assumption that becomes less accurate as smaller scales are included in the measurement. It was recently shown from large samples of N-body simulations that the error bars about the BAO obtained this way are in fact up to 15-20 per cent too small. This important bias has shaken the confidence in the way error bars are calculated, and is motivating developments of analyses pipelines that include non-Gaussian features in the matter density fields. In this thesis, we propose general strategies to incorporate non-Gaussian effects in the context of a survey. After describing the high performance N-body code that we used, we present novel properties of the non-Gaussian uncertainty about the matter power spectrum, and explain how these combine with a general survey selection function. Assuming that the non-Gaussian features that are observed in the simulations correspond to those of Nature, this approach is the first unbiased measurement of the error bar about the power spectrum, which simultaneously removes the undesired bias on the BAO error. We then relax this assumption about the similitude of the non-Gaussian natures in simulations and data, and develop tools that aim at measuring the non-Gaussian error bars exclusively from the data. It is possible to improve the constraining power of non-Gaussian analyses with `Gaussianizations' techniques, which map the observed fields into something more Gaussian. We show that two of such techniques maximally recover degrees of freedom that were lost in the gravitational collapse. Finally, from a large sample of high resolution N-body realizations, we construct a series of weak gravitational lensing distortion maps and provide high resolution halo catalogues that are used by the CFTHLenS community to calibrate their estimators and study many secondary effects with unprecedented accuracy.

Les jeunes amas d'étoiles sont sous-structurés et évoluent dynamiquement pour former des amas sphériques à l'équilibre. Je présente une nouvelle méthode pour générer des conditions initiales réalistes pour simuler ce processus: la fragmentation de Hubble-Lemaitre. Je laisse le système développer spontanément des surdensités au cours d'une expansion du système. Le modèle résultant se compare bien aux simulations hydrodynamiques de formation stellaire et aux observations des jeunes amas. Le modèle fragmenté s'effondre de manière plus douce qu'un modèle uniforme. L'injection d'une population d'étoile binaire avant l'effondrement a montré qu'un système sous-structuré détruisait bien plus de binaires qu'un système à l'équilibre. Des binaires particulièrement larges ou serrées, jusqu’à 0.01 AU, ont également été détectées dans ces modèles. Cette méthode est très prometteuse, un exemple d'application est la génération d'observations synthétiques de régions de formation stellaire. / Young star clusters are substructured and undergo a dynamical evolution erasing this substructure to form relaxed spherical clusters. I present a new method to generate realistic initial conditions to perform N-body simulations of this process: the Hubble-Lemaitre fragmentation. By expanding an initially uniform sphere, I allow spontaneous overdensities to grow, creating a realistic model for young clumpy stellar systems. This method is validated by analysing the distribution and content of the clumps and comparing them to hydrodynamical simulations of star formation as well as observations of star forming regions. These systems undergo a softer collapse than uniform ones. I injected binary stars in the fragmented models and found they were heavily processed when substructure was present. I also found extreme short and tight binaries, down to 0.01 AU, to formin the models. The method has a lot of potential, such as the generation of mock observations of star-forming regions.

The following study examines whether the initialisation (multipole expansions at the finest level) and evaluation of the numerical method Fast Multipole Method (FMM) can benefit from implementing SSE instructions. The implementation of SSE-instructions have been studied and compared to the serial case. Moreover, studied parts of the algorithm include arithmetics on complex numbers, and the usage of applying SSE instructions to complex numbers of double precision. In conclusion, the initialisation has not experienced any improvement in terms of throughput by appliying SSE instructions. However, the evaluation reached almost the double speed-up when SSE instructions were applied. The difference in results are most likely due to the structure of the both algorithms. The initialisation is simple, but the evaluation which involves more operations can benefit from SSE instructions. Furthermore, a scheme is proposed for how SSE instructions can be applied to data sets which are not divisable by the unroll factor and to data sets of varying size.

We study asteroidal breakups, i.e. fragmentations of targets, subsequent gravitational reaccumulation and formation of small asteroid families. We fo- cused on parent bodies with diameters Dpb = 10 km. Simulations were per- formed with a smoothed-particle hydrodynamics (SPH) code combined with an efficient N-body integrator. We assumed various projectile sizes, impact veloci- ties and angles (125 runs in total). Resulting size-frequency distributions are sig- nificantly different from results of scaled-down simulations with Dpb = 100 km targets (Durda et al. 2007). We thus derive new parametric relations describing fragment distributions, suitable for Monte-Carlo collisional models. We also characterize velocity fields and angular distributions of fragments, which can be used in N-body simulations of asteroid families. Finally, we discuss several uncertainties related to SPH simulations.