The Aether-Scalar-Tensor (AeST) theory is an extension of General Relativity (GR) which can support Modified Newtonian Dynamics (MOND) behaviour in its static weak-field limit, and cosmological evolution resembling ΛCDM. We consider static spherically symmetric weak-field solutions in this theory and show that the resulting equations can be reduced to a single equation for the gravitational potential. The reduced equation has apparent isolated singularities at the zeros of the derivative of the potential and we show how these are removed by evolving, instead, the canonical momentum of the corresponding Hamiltonian system that we find. We construct solutions in three cases: (i) in vacuum outside a bounded spherical object, (ii) within an extended prescribed source, and (iii) for an isothermal gas in hydrostatic equilibrium, serving as a simplified model for galaxy clusters. We show that the oscillatory regime that follows the Newtonian and MOND regimes, obtained in previous works in the vacuum case, also persists for isothermal spheres, and we show that the gas density profiles in AeST can become more compressed than their Newtonian or MOND counterparts. We construct the Radial Acceleration Relation (RAR) in AeST for isothermal spheres and find that it can display a peak, an enhancement with respect to the MOND RAR, at an acceleration range determined by the value of the AeST weak-field mass parameter, the mass of the system and the boundary value of the gravitational potential. For lower accelerations, the AeST RAR drops below the MOND expectation, as if there is a negative mass density. Similar observational features of the galaxy cluster RAR have been reported. This illustrates the potential of AeST to address the shortcomings of MOND in galaxy clusters, but a full quantitative comparison with observations will require going beyond the isothermal case.

We have developed a precise dictionary between the spectrum of primordial density fluctuations and the parameters of the effective field theory (EFT) of inflation that determine the primordial power spectrum (PPS). At lowest order the EFT contains two parameters: the slow-roll parameter $\epsilon$, which acts as an order parameter, and the speed of sound $c_s$. Applying second-order perturbation theory, we provide maps from the PPS to the EFT parameters that are precise up to the cube of the fractional change in the PPS $(\Delta \mathcal{P}/\mathcal{P})^3$, or less than $1\%$ for spectral features that modulate the PPS by $20\%$. While such features are not required when the underlying cosmological model is assumed to be $\Lambda$CDM they are necessary for alternative models that have no cosmological constant/dark energy. We verify the dictionary numerically and find those excursions in the slow-roll parameter that reproduce the PPS needed to fit Planck data for both $\Lambda$ and no-$\Lambda$ cosmological models.

Reconstructions of the primordial power spectrum (PPS) of curvature perturbations from cosmic microwave background anisotropies and large-scale structure data suggest that the usually assumed power-law PPS has localised features (up to \sim 10\%∼10% in amplitude), although of only marginal significance in the framework of \LambdaΛCDM cosmology. On the other hand if the cosmology is taken to be Einstein-de Sitter, larger features in the PPS (up to \sim 20\%∼20% in amplitude) are required to accurately fit the observed acoustic peaks. Within the context of single clock inflation, we show that any given reconstruction of the PPS can be mapped on to functional parameters of the underlying effective theory of the adiabatic mode within a 2nd-order formalism, provided the best fit fractional change of the PPS, \Delta{P}_{R}/{P}_{R}ΔPR/PR is such that (\Delta{P}_{R}/{P}_{R})^3(ΔPR/PR)3 falls within the 1\,\sigma1σ confidence interval of the reconstruction for features induced by variations of either the sound speed c_\mathrm{s}cs or the slow-roll parameter \epsilonϵ. Although there is a degeneracy amongst these functional parameters (and the models that project onto them), we can identify simple representative inflationary models that yield such features in the PPS. Thus we provide a dictionary (more accurately, a thesaurus) to go from observational data, via the reconstructed PPS, to models that reproduce them to per cent level precision.

We consider the possibility that the primordial curvature perturbation is direction-dependent. To first order this is parameterised by a quadrupolar modulation of the power spectrum and results in statistical anisotropy of the CMB, which can be quantified using `bipolar spherical harmonics'. We compute these for the Planck DR2-2015 SMICA map and estimate the noise covariance from Planck Full Focal Plane 9 simulations. A constant quadrupolar modulation is detected with 2.2 σ significance, dropping to 2σ when the primordial power is assumed to scale with wave number k as a power law. Going beyond previous work we now allow the spectrum to have arbitrary scale-dependence. Our non-parametric reconstruction then suggests several spectral features, the most prominent at k ∼ 0.006 Mpc−1. When a constant quadrupolar modulation is fitted to data in the range 0.005 ⩽ k/Mpc−1 ⩽ 0.008, its preferred directions are found to be related to the cosmic hemispherical asymmetry and the CMB dipole. To determine the significance we apply two test statistics to our reconstructions of the quadrupolar modulation from data, against reconstructions of realisations of noise only. With a test statistic sensitive only to the amplitude of the modulation, the reconstructions from the multipole range 30 ⩽ ℓ ⩽ 1200 are unusual with 2.1σ significance. With the second test statistic, sensitive also to the direction, the significance rises to 6.9σ. Our approach is easily generalised to include other data sets such as polarisation, large-scale structure and forthcoming 21-cm line observations which will enable these anomalies to be investigated further.

In stochastic quantisation, quantum mechanical expectation values are computed as averages over the time history of a stochastic process described by a Langevin equation. Complex stochastic quantisation, though theoretically not rigorously established, extends this idea to cases where the action is complex-valued by complexifying the basic degrees of freedom, all observables and allowing the stochastic process to probe the complexified configuration space. We review the method for a previously studied one-dimensional toy model, the U(1) one link model. We confirm that complex Langevin dynamics only works for a certain range of parameters, misestimating observables otherwise. A curious effect is observed where all moments of the basic stochastic variable are misestimated, although these misestimated moments may be used to construct, by a Taylor series, other observables that are reproduced correctly. This suggests a subtle but not completely resolved relationship between the original complex integration measure and the higher-dimensional probability distribution in the complexified configuration space, generated by the complex Langevin process.

Of the four exclusive normed division algebras, only the real and complex numbers prevail in both mathematics and physics. The noncommutative quaternions and the nonassociative octonions have found limited physical applications. In mathematics, division algebras unify both classical and exceptional Lie algebras with the exceptional ones appearing in a table known as the magic square generated by tensor products of division algebras. This work reviews the normed division algebras and the magic square as well as necessary preliminaries for its construction. Space-time transformations, pure super Yang-Mills theories in space-time dimensions D = 3, 4, 6, 10, dimensional reduction and truncation of supersymmetry are also described here by the four division algebras. Supergravity theories, seen as tensor products of super Yang-Mills theories, are described as tensor products of division algebras leading to the identi cation of a magic square of supergravity theories with their U-duality groups as the magic square entries, providing applications of all division algebras to physics and suggesting division algebraic underpinnings of supersymmetry. Other curious uses of octonions are also mentioned.