We present three nonparametric Bayesian primordial reconstructions using Planck 2018 polarization data: linear spline primordial power spectrum reconstructions, cubic spline inflationary potential reconstructions, and sharp-featured primordial power spectrum reconstructions. All three methods conditionally show hints of an oscillatory feature in the primordial power spectrum in the multipole range $\ensuremath{\ell}\ensuremath{\sim}20$ to $\ensuremath{\ell}\ensuremath{\sim}50$, which is to some extent preserved upon marginalization. We find no evidence for deviations from a pure power law across a broad observable window ($50\ensuremath{\lesssim}\ensuremath{\ell}\ensuremath{\lesssim}2000$), but find that parametrizations are preferred which are able to account for lack of resolution at large angular scales due to cosmic variance, and at small angular scales due to Planck instrument noise. Furthermore, the late-time cosmological parameters are unperturbed by these extensions to the primordial power spectrum. This work is intended to provide a background and give more details of the Bayesian primordial reconstruction work found in the Planck 2018 papers.
Spherically symmetric Einstein-{\ae}ther (E{\AE}) theory with a Maxwell-like kinetic term is revisited. We consider a general choice of the metric and the \ae{}ther field, finding that:~(i) there is a gauge freedom allowing one always to use a diagonal metric; and~(ii) the nature of the Maxwell equation forces the \ae{}ther field to be time-like in the coordinate basis. We derive the vacuum solution and confirm that the innermost stable circular orbit (ISCO) and photon ring are enlarged relative to general relativity (GR). Buchdahl's theorem in E\AE{} theory is derived. For a uniform physical density, we find that the upper bound on compactness is always lower than in GR. Additionally, we observe that the Newtonian and E\AE{} radial acceleration relations run parallel in the low pressure limit. Our analysis of E\AE{} theory may offer novel insights into its interesting phenomenological generalization: \AE{}ther--scalar--tensor theory ({\AE}ST).
We present 13.9-18.2 GHz observations of the Sunyaev-Zel'dovich (SZ) effect towards Abell 2146 using the Arcminute Microkelvin Imager (AMI). The cluster is detected with a peak SNR ratio of 13 sigma in the radio source subtracted map. Comparison of the SZ and X-ray images suggests that they both have extended regions which lie approximately perpendicular to one another, with their emission peaks significantly displaced. These features indicate non-uniformities in the distributions of the gas temperature and pressure, indicative of a cluster merger. We use a Bayesian cluster analysis to explore the high-dimensional parameter space of the cluster-plus-sources model to obtain cluster parameter estimates in the presence of radio point sources, receiver noise and primordial CMB anisotropy; the probability of SZ + CMB primordial structure + radio sources + receiver noise to CMB + radio sources + receiver noise is 3 x 10^{6}:1. We compare the results from three different cluster models. Our preferred model exploits the observation that the gas fractions do not appear to vary greatly between clusters. Given the relative masses of the two merging systems in Abell 2146, the mean gas temperature can be deduced from the virial theorem (assuming all of the kinetic energy is in the form of internal gas energy) without being affected significantly by the merger event, provided the primary cluster was virialized before the merger. In this model we fit a simple spherical isothermal beta-model, despite the inadequacy of this model for a merging system like Abell 2146, and assume the cluster follows the mass-temperature relation of a virialized, singular, isothermal sphere. We note that this model avoids inferring large-scale cluster parameters internal to r_200 under the widely used assumption of hydrostatic equilibrium. We find that at r_200 M_T= 4.1 \pm 0.5 x 10^{14} h^{-1}M_sun and T=4.5 \pm 0.5 keV.
In the preceding three chapters, we have considered in some detail the Schwarzschild geometry, which represents the gravitational field outside a static spherically symmetric object. We also considered the structure of the Schwarzschild black hole, in which the empty-space field equations are satisfied everywhere except at the central intrinsic singularity. In this chapter, we consider solving the Einstein equations for a static spherically symmetric spacetime in regions where the presence of other fields means that the energy–momentum tensor is non-zero. In particular, we will concentrate on two physically interesting situations. First, we discuss the relativistic gravitational equations for the interior of a spherically symmetric matter distribution (or star); in this case the energy–momentum tensor of the matter making up the star must be included in the Einstein field equations. Second, we consider the spacetime geometry outside a static spherically symmetric charged object; once again this is not a vacuum, since it is filled with a static electric field whose energy–momentum must be included in the field equations.
A stress-energy tensor, τab, for linear gravity in the physical spacetime, M, approximated by a flat background, M̌, and adapted to the harmonic gauge, was recently proposed by Butcher, Hobson, and Lasenby. By removing gauge constraints and imposing full metrical general relativity, we find a natural generalization of τab to the pseudotensor of Einstein, Etab. Møller’s pseudotensor, Mtab, is an alternative to Etab formulated using tetrads and is thus naturally adapted to, e.g., Einstein-Cartan gravity. Gauge theory gravity uses the geometric algebra to reproduce Einstein-Cartan gravity and is a Poincaré gauge theory for the spacetime algebra: the tetrad and spin connection appear as gauge fields on Minkowski space, M4. We obtain the pseudotensor of Møller for gauge theory gravity, Mt(a), using a variational approach, also identifying a potentially interesting recipe for constructing conserved currents in that theory. We show that in static, spherical spacetimes containing a gravitational mass MT, the pseudotensors in the spacetime algebra, Mt(a) and Et(a), describe gravitational stress-energy as if the gravitational potential were a scalar (i.e., Klein-Gordon) field, φ, coupled to gravitational mass density, ϱ, on the Minkowski background M4. The old Newtonian formula φ = −MT/r successfully describes even strong fields in this picture. The Newtonian limit of this effect was previously observed in τab on M̌ for linear gravity. We also draw fresh attention to the conserved mass of a static system, MTMT. We observe that the gravitational energy of Einstein and Møller was added to MT on M4 to give MT. We demonstrate the Klein-Gordon correspondence and mass functions using the “Schwarzschild star” solution for an incompressible perfect fluid ball.
We describe a theoretical procedure for analyzing astronomical phased arrays with overlapping beams and apply the procedure to simulate a simple example. We demonstrate the effect of overlapping beams on the number of degrees of freedom of the array and on the ability of the array to recover a source. We show that the best images are obtained using overlapping beams, contrary to common practice, and show how the dynamic range of a phased array directly affects the image quality.
It was recently found that, when linearised in the absence of matter, 58 cases of the general gravitational theory with quadratic curvature and torsion are (i) free from ghosts and tachyons and (ii) power-counting renormalisable. We inspect the nonlinear Hamiltonian structure of the eight cases whose primary constraints do not depend on the curvature tensor. We confirm the particle spectra and unitarity of all these theories in the linear regime. We uncover qualitative dynamical changes in the nonlinear regimes of all eight cases, suggesting at least a broken gauge symmetry, and possibly the activation of negative kinetic energy spin-parity sectors and acausal behaviour. Two of the cases propagate a pair of massless modes at the linear level, and were interesting as candidate theories of gravity. However, we identify these modes with vector excitations, rather than the tensor polarisations of the graviton. Moreover, we show that these theories do not support a viable cosmological background.
We introduce dynamic nested sampling: a generalisation of the nested sampling algorithm in which the number of "live points" varies to allocate samples more efficiently. In empirical tests the new method significantly improves calculation accuracy compared to standard nested sampling with the same number of samples; this increase in accuracy is equivalent to speeding up the computation by factors of up to ~72 for parameter estimation and ~7 for evidence calculations. We also show that the accuracy of both parameter estimation and evidence calculations can be improved simultaneously. In addition, unlike in standard nested sampling, more accurate results can be obtained by continuing the calculation for longer. Popular standard nested sampling implementations can be easily adapted to perform dynamic nested sampling, and several dynamic nested sampling software packages are now publicly available.
Bayesian inference involves two main computational challenges. First, in estimating the parameters of some model for the data, the posterior distribution may well be highly multi-modal: a regime in which the convergence to stationarity of traditional Markov Chain Monte Carlo (MCMC) techniques becomes incredibly slow. Second, in selecting between a set of competing models the necessary estimation of the Bayesian evidence for each is, by definition, a (possibly high-dimensional) integration over the entire parameter space; again this can be a daunting computational task, although new Monte Carlo (MC) integration algorithms offer solutions of ever increasing efficiency. Nested sampling (NS) is one such contemporary MC strategy targeted at calculation of the Bayesian evidence, but which also enables posterior inference as a by-product, thereby allowing simultaneous parameter estimation and model selection. The widely-used MultiNest algorithm presents a particularly efficient implementation of the NS technique for multi-modal posteriors. In this paper we discuss importance nested sampling (INS), an alternative summation of the MultiNest draws, which can calculate the Bayesian evidence at up to an order of magnitude higher accuracy than `vanilla' NS with no change in the way MultiNest explores the parameter space. This is accomplished by treating as a (pseudo-)importance sample the totality of points collected by MultiNest, including those previously discarded under the constrained likelihood sampling of the NS algorithm. We apply this technique to several challenging test problems and compare the accuracy of Bayesian evidences obtained with INS against those from vanilla NS.
'%3e%3cpath fill='%23012169' d='M0 0v30h60V0z'/%3e%3cpath stroke='%23fff' stroke-width='6' d='m0 0 60 30m0-30L0 30'/%3e%3cpath stroke='%23C8102E' stroke-width='4' d='m0 0 60 30m0-30L0 30' clip-path='url(%23b)'/%3e%3cpath stroke='%23fff' stroke-width='10' d='M30 0v30M0 15h60'/%3e%3cpath stroke='%23C8102E' stroke-width='6' d='M30 0v30M0 15h60'/%3e%3c/g%3e%3c/svg%3e)
