Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with ...early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.
polychord: nested sampling for cosmology Handley, W. J; Hobson, M. P; Lasenby, A. N
Monthly notices of the Royal Astronomical Society. Letters,
06/2015, Volume:
450, Issue:
1
Journal Article
Peer reviewed
Open access
polychord is a novel nested sampling algorithm tailored for high-dimensional parameter spaces. In addition, it can fully exploit a hierarchy of parameter speeds such as is found in cosmomc and camb. ...It utilizes slice sampling at each iteration to sample within the hard likelihood constraint of nested sampling. It can identify and evolve separate modes of a posterior semi-independently and is parallelized using openmpi. polychord is available for download at http://ccpforge.cse.rl.ac.uk/gf/project/polychord/.
polychord: next-generation nested sampling Handley, W. J; Hobson, M. P; Lasenby, A. N
Monthly notices of the Royal Astronomical Society,
11/2015, Volume:
453, Issue:
4
Journal Article
Peer reviewed
Open access
polychord is a novel nested sampling algorithm tailored for high-dimensional parameter spaces. This paper coincides with the release of polychord v1.6, and provides an extensive account of the ...algorithm. polychord utilizes slice sampling at each iteration to sample within the hard likelihood constraint of nested sampling. It can identify and evolve separate modes of a posterior semi-independently, and is parallelized using openmpi. It is capable of exploiting a hierarchy of parameter speeds such as those present in cosmomc and camb, and is now in use in the cosmochord and modechord codes. polychord is available for download from http://ccpforge.cse.rl.ac.uk/gf/project/polychord/.
Abstract
Data-driven model-independent reconstructions of the dark energy equation of state w(z) are presented using Planck 2015 era cosmic microwave background, baryonic acoustic oscillations (BAO), ...Type Ia supernova (SNIa) and Lyman α (Lyα) data. These reconstructions identify the w(z) behaviour supported by the data and show a bifurcation of the equation of state posterior in the range 1.5 < z < 3. Although the concordance Λ cold dark matter (ΛCDM) model is consistent with the data at all redshifts in one of the bifurcated spaces, in the other, a supernegative equation of state (also known as ‘phantom dark energy’) is identified within the 1.5σ confidence intervals of the posterior distribution. To identify the power of different data sets in constraining the dark energy equation of state, we use a novel formulation of the Kullback–Leibler divergence. This formalism quantifies the information the data add when moving from priors to posteriors for each possible data set combination. The SNIa and BAO data sets are shown to provide much more constraining power in comparison to the Lyα data sets. Further, SNIa and BAO constrain most strongly around redshift range 0.1–0.5, whilst the Lyα data constrain weakly over a broader range. We do not attribute the supernegative favouring to any particular data set, and note that the ΛCDM model was favoured at more than 2 log-units in Bayes factors over all the models tested despite the weakly preferred w(z) structure in the data.
A method is presented for Bayesian model selection without explicitly computing evidences, by using a combined likelihood and introducing an integer model selection parameter n so that Bayes factors, ...or more generally posterior odds ratios, may be read off directly from the posterior of n. If the total number of models under consideration is specified a priori, the full joint parameter space (θ, n) of the models is of fixed dimensionality and can be explored using standard Markov chain Monte Carlo (MCMC) or nested sampling methods, without the need for reversible jump MCMC techniques. The posterior on n is then obtained by straightforward marginalization. We demonstrate the efficacy of our approach by application to several toy models. We then apply it to constraining the dark energy equation of state using a free-form reconstruction technique. We show that Λ cold dark matter is significantly favoured over all extensions, including the simple w(z) = constant model.
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