On the Centre Manifold of Collinear Points in the Planar Three-Body Problem

We study the existence of invariant tori in a neighbourhood of the collinear equilibrium points of the planar three-body problem. To this end some properties of the normal form of the Hamiltonian reduced to the 4D central manifold are proved. Using this normal form, we show that the nondegeneracy conditions of KAM theorem are satisfied for all positive masses, including the 2:1 resonance case. The evaluation of the conditions is done numerically.     

Large-scale structure, the cosmic microwave background and primordial non-Gaussianity

Cosmic microwave background and large-scale structure data will shortly improve dramatically with the Microwave Anisotropy Probe and Planck Surveyor , and the Anglo-Australian 2-Degree Field and Sloan Digital Sky Survey. It is therefore timely to ask which of the microwave background and large-scale structure will provide a better probe of primordial non-Gaussianity. In this paper we consider this question, using the bispectrum as a discriminating statistic. We consider several non-Gaussian models and find that in each case the microwave background will provide a better probe of primordial non-Gaussianity. Our results suggest that if microwave background maps appear Gaussian, then apparent deviations from Gaussian initial conditions in galaxy surveys can be attributed with confidence to the effects of biasing. We demonstrate this precisely for the spatial bispectrum induced by local non-linear biasing.     

Large-scale cosmic homogeneity from a multifractal analysis of the PSCz catalogue

We investigate the behaviour of galaxy clustering on large scales using the PSC z catalogue. In particular, we ask whether there is any evidence of large-scale fractal behaviour in this catalogue. We find the correlation dimension in this survey varies with scale, consistent with other analyses. For example, our results on small and intermediate scales are consistent those obtained from the QDOT sample, but the larger PSC z sample allows us to extend the analysis out to much larger scales. We find firm evidence that the sample becomes homogeneous at large scales; the correlation dimension of the sample is D ₂=2.992±0.003 for r >30  h ⁻¹ Mpc. This provides strong evidence in favour of a universe that obeys the cosmological principle.     

A Study on the Light Curves of GRBs

It is generally believed that the complexity and variability of the light curves of gamma-ray bursts (GRBs) are caused by the internal shocks, which would occur when a rapid shell catches up a slower one and collides with it. The electrons in the shock layer are heated by the shocks and radiate via the mechanisms of synchrotron radiation and inverse Compton scattering. Based on relativistic kinematics, a relation between the photon number of the emission from the rapidly moving shock layer and the number of the photons received by an observer is derived. Then, employing the angular spreading of the internal shock emission, the curve equation and profile of a single pulse are obtained, and the shape is typically in the shape of a fast rise and exponential decline. Furthermore, by using the model of the successive collisions of multiple shells under the condition of reasonable parameters, the observed light curves are fitted with a rather good effect. Therefore, by this means, more different types of light curves of GRBs can be explained.