A Fast Parameterized Algorithm for Co-Path Set

The k-CO-PATH SET problem asks, given a graph G and a positive integer k, whether one can delete k edges from G so that the remainder is a collection of disjoint paths. We give a linear-time fpt algorithm with complexity O^*(1.588^k) for deciding k-CO-PATH SET, significantly improving the previously best known O^*(2.17^k) of Feng, Zhou, and Wang (2015). Our main tool is a new O^*(4^{tw(G)}) algorithm for CO-PATH SET using the Cut&Count framework, where tw(G) denotes treewidth. In general graphs, we combine this with a branching algorithm which refines a 6k-kernel into reduced instances, which we prove have bounded treewidth

Cosmic structure formation in Hybrid Inflation models

A wide class of inflationary models, known as Hybrid Inflation models, may produce topological defects during a phase transition at the end of the inflationary epoch. We point out that, if the energy scale of these defects is close to that of Grand Unification, then their effect on cosmic structure formation and the generation of microwave background anisotropies cannot be ignored. Therefore, it is possible for structure to be seeded by a combination of the adiabatic perturbations produced during inflation and active isocurvature perturbations produced by defects. Since the two mechanisms are uncorrelated the power spectra can be computed by a weighted average of the individual contributions. We investigate the possible observational consequences of this with reference to general Hybrid Inflation models and also a specific model based on Supergravity. These mixed perturbation scenarios have some novel observational consequences and these are discussed qualitatively.Comment: 22 Page

Towards models of gravitational waveforms from generic binaries: A simple approximate mapping between precessing and non-precessing inspiral signals

One of the greatest theoretical challenges in the build-up to the era of second-generation gravitational-wave detectors is the modeling of generic binary waveforms. We introduce an approximation that has the potential to significantly simplify this problem. We show that generic precessing-binary inspiral waveforms (covering a seven-dimensional space of intrinsic parameters) can be mapped to a two-dimensional space of non-precessing binaries, characterized by the mass ratio and a single effective total spin. The mapping consists of a time-dependent rotation of the waveforms into the quadrupole-aligned frame, and is extremely accurate (matches $> 0.99$ with parameter biases in the total spin of $\Delta \chi \leq 0.04$), even in the case of transitional precession. In addition, we demonstrate a simple method to construct hybrid post-Newtonian--numerical-relativity precessing-binary waveforms in the quadrupole-aligned frame, and provide evidence that our approximate mapping can be used all the way to the merger. Finally, based on these results, we outline a general proposal for the construction of generic waveform models, which will be the focus of future work.Comment: 16 pages, 11 figures, 2 tables; replaced to match published version; journal ref. adde

The Cosmic Microwave Background and Particle Physics

In forthcoming years, connections between cosmology and particle physics will be made increasingly important with the advent of a new generation of cosmic microwave background (CMB) experiments. Here, we review a number of these links. Our primary focus is on new CMB tests of inflation. We explain how the inflationary predictions for the geometry of the Universe and primordial density perturbations will be tested by CMB temperature fluctuations, and how the gravitational waves predicted by inflation can be pursued with the CMB polarization. The CMB signatures of topological defects and primordial magnetic fields from cosmological phase transitions are also discussed. Furthermore, we review current and future CMB constraints on various types of dark matter (e.g. massive neutrinos, weakly interacting massive particles, axions, vacuum energy), decaying particles, the baryon asymmetry of the Universe, ultra-high-energy cosmic rays, exotic cosmological topologies, and other new physics.Comment: 43 pages. To appear in Annual Reviews of Nuclear and Particle Scienc

A Bayesian Approach to the Detection Problem in Gravitational Wave Astronomy

The analysis of data from gravitational wave detectors can be divided into three phases: search, characterization, and evaluation. The evaluation of the detection - determining whether a candidate event is astrophysical in origin or some artifact created by instrument noise - is a crucial step in the analysis. The on-going analyses of data from ground based detectors employ a frequentist approach to the detection problem. A detection statistic is chosen, for which background levels and detection efficiencies are estimated from Monte Carlo studies. This approach frames the detection problem in terms of an infinite collection of trials, with the actual measurement corresponding to some realization of this hypothetical set. Here we explore an alternative, Bayesian approach to the detection problem, that considers prior information and the actual data in hand. Our particular focus is on the computational techniques used to implement the Bayesian analysis. We find that the Parallel Tempered Markov Chain Monte Carlo (PTMCMC) algorithm is able to address all three phases of the anaylsis in a coherent framework. The signals are found by locating the posterior modes, the model parameters are characterized by mapping out the joint posterior distribution, and finally, the model evidence is computed by thermodynamic integration. As a demonstration, we consider the detection problem of selecting between models describing the data as instrument noise, or instrument noise plus the signal from a single compact galactic binary. The evidence ratios, or Bayes factors, computed by the PTMCMC algorithm are found to be in close agreement with those computed using a Reversible Jump Markov Chain Monte Carlo algorithm.Comment: 19 pages, 12 figures, revised to address referee's comment