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, Letnik:
450, Številka:
1
Journal Article
Recenzirano
Odprti dostop
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, Letnik:
453, Številka:
4
Journal Article
Recenzirano
Odprti dostop
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/.
We consider Weyl gauge theories of gravity (WGTs), which are invariant both under local Poincaré transformations and local changes of scale. Such theories may be interpreted as gauge theories in ...Minkowski spacetime, but their gravitational interactions are most often reinterpreted geometrically in terms of a Weyl-Cartan spacetime, in which any matter fields then reside. Such a spacetime is a straightforward generalization of Weyl spacetime to include torsion. As first suggested by Einstein, Weyl spacetime is believed to exhibit a so-called second clock effect, which prevents the existence of experimentally observed sharp spectral lines, since the rates of (atomic) clocks depend on their past history. The prevailing view in the literature is that this rules out WGTs as unphysical. Contrary to this viewpoint, we show that if one adopts the natural covariant derivative identified in the geometric interpretation of WGTs, properly takes into account the scaling dimension of physical quantities, and recognizes that Einstein's original objection requires the presence of massive matter fields to represent atoms, observers and clocks, then WGTs do not predict a second clock effect.
We reconsider the widely held view that the Mannheim–Kazanas (MK) vacuum solution for a static, spherically symmetric system in conformal gravity (CG) predicts flat rotation curves, such as those ...observed in galaxies, without the need for dark matter. This prediction assumes that test particles have fixed rest mass and follow timelike geodesics in the MK metric in the vacuum region exterior to a spherically symmetric representation of the galactic mass distribution. Such geodesics are not conformally invariant, however, which leads to an apparent discrepancy with the analogous calculation performed in the conformally equivalent Schwarzschild–de-Sitter (SdS) metric, where the latter does not predict flat rotation curves. This difference arises since the mass of particles in CG must instead be generated dynamically through interaction with a scalar field. The energy-momentum of this required scalar field means that, in a general conformal frame from the equivalence class of CG solutions outside a static, spherically symmetric matter distribution, the spacetime is not given by the MK vacuum solution. A unique frame does exist, however, for which the metric retains the MK form, since the scalar field energy-momentum vanishes despite the field being nonzero and radially dependent. Nonetheless, we show that in both this MK frame and the Einstein frame, in which the scalar field is constant, massive particles follow timelike geodesics of the SdS metric, thereby resolving the apparent frame dependence of physical predictions and unambiguously yielding rotation curves with no flat region. Moreover, we show that attempts to model rising rotation curves by fitting the coefficient of the quadratic term in the SdS metric individually for each galaxy are also precluded, since the scalar field equation of motion introduces an additional constraint relative to the vacuum case, such that the coefficient of the quadratic term in the SdS metric is most naturally interpreted as proportional to a global cosmological constant. We also comment briefly on how our analysis resolves the long-standing uncertainty regarding gravitational lensing in the MK metric.
We review the effect that the choice of a uniform or logarithmic prior has on the Bayesian evidence and hence on Bayesian model comparisons when data provide only a one-sided bound on a parameter. We ...investigate two particular examples: the tensor-to-scalar ratio r of primordial perturbations and the mass of individual neutrinos mν, using the cosmic microwave background temperature and polarization data from Planck 2018 and the NuFIT 5.0 data from neutrino oscillation experiments. We argue that the Kullback–Leibler divergence, also called the relative entropy, mathematically quantifies the Occam penalty. We further show how the Bayesian evidence stays invariant upon changing the lower prior bound of an upper constrained parameter. While a uniform prior on the tensor-to-scalar ratio disfavors the r extension compared to the base ΛCDM model with odds of about 1∶20, switching to a logarithmic prior renders both models essentially equally likely. ΛCDM with a single massive neutrino is favored over an extension with variable neutrino masses with odds of 20∶1 in case of a uniform prior on the lightest neutrino mass, which decreases to roughly 2∶1 for a logarithmic prior. For both prior options we get only a very slight preference for the normal over the inverted neutrino hierarchy with Bayesian odds of about 3∶2 at most.
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.
We present cosmological constraints from Planck 2015 data for a Universe that is kinetically dominated at very early times. We perform a Markov chain Monte Carlo analysis to estimate parameters and ...use nested sampling to determine the evidence for a model comparison of the single-field quadratic and Starobinsky inflationary models with the standard ΛCDM cosmology. In particular we investigate how different amounts of inflation before and after horizon exit affect the primordial power spectrum and subsequently the power spectrum of the cosmic microwave background. We find that the model using kinetically dominated initial conditions for inflation performs similarly well in terms of Bayesian evidence as a model directly starting out in the slow-roll phase, despite having an additional parameter. The data show a slight preference for a cutoff at large scales in the primordial and temperature power spectra.
We consider the construction of gauge theories of gravity that are invariant under local conformal transformations. We first clarify the geometric nature of global conformal transformations, in both ...their infinitesimal and finite forms, and the consequences of global conformal invariance for field theories, before reconsidering existing approaches for gauging the conformal group, namely auxiliary conformal gauge theory and biconformal gauge theory, neither of which is generally accepted as a complete solution. We then demonstrate that, provided any matter fields belong to an irreducible representation of the Lorentz group, the recently proposed extended Weyl gauge theory (eWGT) may be considered as an alternative method for gauging the conformal group, since eWGT is invariant under the full set of local conformal transformations, including inversions, as well as possessing conservation laws that provide a natural local generalization of those satisfied by field theories with global conformal invariance, and also having an "ungauged" limit that corresponds to global conformal transformations. By contrast, although standard Weyl gauge theory also enjoys the first of these properties, it does not share the other two, and so cannot be considered a valid gauge theory of the conformal group.
We make a case for setting initial conditions for inflation at the Planck epoch in the kinetically dominated regime. For inflationary potentials with a plateau or a hill, i.e., potentials that are ...bounded from above within a certain region of interest, we cannot claim complete ignorance of the energy distribution between kinetic and potential energy, and equipartition of energy at the Planck epoch becomes questionable. We analyze different classes of potentials in phase space and quantify the fraction of the Planck surface that is kinetically dominated. Considering bounded potentials with very small amplitudes as favored by current data and restricting ourselves to the domains of phase space that are of interest to cosmic inflation, we find that initial conditions of the inflaton field should be set in the kinetically dominated regime regardless of any choice of prior.
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