Search

We propose a new theory of dark matter based on axion physics and cosmological phase transitions. We show that theories in which a gauge coupling increases through a first-order phase transition naturally result in `axion relic pockets': regions of relic false vacua stabilised by the pressure from a kinematically trapped, hot axion gas. Axion relic pockets provide a viable and highly economical theory of dark matter: the macroscopic properties of the pockets depend only on a single parameter (the phase transition temperature). We describe the formation, evolution and present-day properties of axion relic pockets, and outline how their phenomenology is distinct from existing dark matter paradigms. We briefly discuss how laboratory experiments and astronomical observations can be used to test the theory, and identify gamma-ray observations of magnetised, dark-matter-dense environments as particularly promising., Comment: 22 pages, 6 figures. v2: References and a figure added, typos corrected. Matches published version

In this paper, we explore the phenomenology of massive Axion-Like Particles (ALPs) coupled to quarks and gluons, dubbed 'QCD ALPs', with an emphasis on the associated low-energy observables. ALPs coupled to gluons and quarks not only induce nuclear interactions at scales below the QCD-scale, relevant for ALP production in supernovae (SNe), but naturally also couple to photons similarly to the QCD-axion. We discuss the link between the high-energy formulation of ALP theories and their effective couplings with nucleons and photons. The induced photon coupling allows ALPs with masses $m_a\gtrsim1$ MeV to efficiently decay into photons, and astrophysical observables severely constrain the ALP parameter space. We show that a combination of arguments related to SN events rule out ALP-nucleon couplings down to $g_{aN}\gtrsim 10^{-11}- 10^{-10}$ for $m_a\gtrsim1$ MeV - a region of the parameter space that was hitherto unconstrained., Comment: 15 pages, 3 figures. Matches the version published on PRD. Changes: we added a further discussion about uncertainties related to the Supernova model employed

We investigate the characteristics of the gamma-ray signal following the decay of MeV-scale Axion-Like Particles (ALPs) coupled to photons which are produced in a Supernova (SN) explosion. This analysis is the first to include the production of heavier ALPs through the photon coalescence process, enlarging the mass range of ALPs that could be observed in this way and giving a stronger bound from the observation of SN 1987A. Furthermore, we present a new analytical method for calculating the predicted gamma-ray signal from ALP decays. With this method we can rigorously prove the validity of an approximation that has been used in some of the previous literature, which we show here to be valid only if all gamma rays arrive under extremely small observation angles (i.e. very close to the line of sight to the SN). However, it also shows where the approximation is not valid, and offers an efficient alternative to calculate the ALP-induced gamma-ray flux in a general setting when the observation angles are not guaranteed to be small. We also estimate the sensitivity of the Fermi Large Area Telescope (Fermi-LAT) to this gamma-ray signal from a future nearby SN and show that in the case of a non-observation the current bounds on the ALP-photon coupling $ g_{a\gamma} $ are strengthened by about an order of magnitude. In the case of an observation, we show that it may be possible to reconstruct the product $ g_{a\gamma}^2 m_a $, with $ m_a $ the mass of the ALP., Comment: 29 pages, 9 figures; v3: matches published version

We show that theories of inflation with multiple, rapidly turning fields can generate large amounts of non-Gaussianity. We consider a general theory with two fields, an arbitrary field-space metric, and a potential that supports sustained, rapidly turning field trajectories. Our analysis accounts for non-zero field cross-correlation and does not fix the power spectra of curvature and isocurvature perturbations to be equal at horizon crossing. Using the $\delta N$ formalism, we derive a novel, analytical formula for bispectrum generated from multi-field mixing on super-horizon scales. Rapid-turn inflation can produce a bispectrum with several potentially large contributions that are not necessarily of the local shape. We exemplify the applicability of our formula with a fully explicit model and show that the new contributions indeed can generate a large amplitude of local non-Gaussianity, $f_{\rm NL}^{\rm loc}\sim {\cal O}(1)$. These results will be important when interpreting the outcomes of future observations., Comment: 33 pages, 9 figures

Axions and photons resonantly interconvert in regions where the plasma frequency approximately equals the axion mass. This process is directly analogous to an avoided level crossing in quantum mechanics, for which the celebrated Landau-Zener (LZ) formula provides a simple, non-perturbative solution for the conversion probability. The LZ formula is commonly used in studies of axion-photon conversion; however, in this context, it relies on the assumption that the magnetic field variation is small compared to variations of the plasma frequency, which is frequently not the case in real plasmas. We derive a generalised version of the LZ formula by allowing the boundaries to be located at a scale that is similar to the inhomogeneities. We find that the LZ formula fails when the oscillation range is small compared to the resonance region. This failure is more severe in the adiabatic limit, when the plasma frequency varies slowly, resulting in a conversion probability that is not maximal as opposed to a naive application of the LZ formula. Moreover, we consider circumstances where the generalised LZ formula does not apply and present an alternative semi-classical approximation with complementary regime of validity., Comment: v2: 19 pages, 5 figures. minor changes to match the published version

Gravitationally bound clumps of dark matter axions in the form of 'miniclusters' or even denser 'axion stars' can generate strong radio signals through axion-photon conversion when encountering highly magnetised neutron star magnetospheres. We systematically study encounters of axion clumps with neutron stars and characterise the axion infall, conversion and the subsequent propagation of the photons. We show that the high density and low escape velocity of the axion clumps lead to strong, narrow, and temporally characteristic transient radio lines with an expected duration varying from seconds to months. Our work comprises the first end-to-end modeling pipeline capable of characterizing the radio signal generated during these transient encounters, quantifying the typical brightness, anisotropy, spectral width, and temporal evolution of the radio flux. The methods developed here may prove essential in developing dedicated radio searches for transient radio lines arising from miniclusters and axion stars., Comment: v2: Minor updates made to match published version. v1: 17 pages, 15 figures

Axion-Like Particles (ALPs) are well-motivated extensions of the Standard Model of Particle Physics and a generic prediction of some string theories. X-ray observations of bright Active Galactic Nuclei (AGN) hosted by rich clusters of galaxies are excellent probes of very-light ALPs, with masses $\mathrm{log}(m_\mathrm{a}/\mathrm{eV}) < -12.0$. We evaluate the potential of future X-ray observatories, particularly $Athena$ and the proposed $AXIS$, to constrain ALPs via observations of cluster-hosted AGN, taking NGC 1275 in the Perseus cluster as our exemplar. Assuming perfect knowledge of instrument calibration, we show that a modest exposure (200-ks) of NGC 1275 by $Athena$ permits us to exclude all photon-ALP couplings $g_\mathrm{a\gamma} > 6.3 \times 10^{-14} \ {\mathrm{GeV}}^{-1}$ at the 95% level, as previously shown by $Conlon \ et \ al. \ (2018)$, representing a factor of 10 improvement over current limits. We then proceed to assess the impact of realistic calibration uncertainties on the $Athena$ projection by applying a standard $Cash$ likelihood procedure, showing the projected constraints on $g_\mathrm{a\gamma}$ weaken by a factor of 10 (back to the current most sensitive constraints). However, we show how the use of a deep neural network can disentangle the energy-dependent features induced by instrumental miscalibration and those induced by photon-ALP mixing, allowing us to recover most of the sensitivity to the ALP physics. In our explicit demonstration, the machine learning applied allows us to exclude $g_\mathrm{a\gamma} > 2.0 \times 10^{-13} \ {\mathrm{GeV}}^{-1}$, complementing the projected constraints of next-generation ALP dark matter birefringent cavity searches for very-light ALPs. Finally, we show that a 200-ks $AXIS$/on-axis observation of NGC 1275 will tighten the current best constraints on very-light ALPs by a factor of 3., Comment: 22 pages, 10 figures, 3 tables. Accepted for publication at the Astrophysical Journal. Comments welcome

The extraordinarily bright gamma-ray burst GRB 221009A was observed by a large number of observatories, from radio frequencies to gamma-rays. Of particular interest are the reported observations of photon-like air showers of very high energy: an 18 TeV event in LHAASO and a 251 TeV event at Carpet-2. Gamma rays at these energies are expected to be absorbed by pair-production events on background photons when travelling intergalactic distances. Several works have sought to explain the observations of these events, assuming they originate from GRB 221009A, by invoking axion-like particles (ALPs). We reconsider this scenario and account for astrophysical uncertainties due to poorly known magnetic fields and background photon densities. We find that, robustly, the ALP scenario cannot simultaneously account for an 18 TeV and a 251 TeV photon from GRB 221009A., Comment: 16 pages, 6 figures. v2: major changes to fig. 5, conclusions changed accordingly

The interconversion of axionlike particles (ALPs) and photons in magnetised astrophysical environments provides a promising route to search for ALPs. The strongest limits to date on light ALPs use galaxy clusters as ALP-photon converters. However, such studies traditionally rely on simple models of the cluster magnetic fields, with the state-of-the-art being Gaussian random fields (GRFs). We present the first systematic study of ALP-photon conversion in more realistic, turbulent fields from dedicated magnetohydrodynamic (MHD) simulations, which we compare with GRF models. For GRFs, we analytically derive the distribution of conversion ratios at fixed energy and find that it follows an exponential law. We find that the MHD models agree with the exponential law for typical, small-amplitude mixings but exhibit distinctly heavy tails for rare and large mixings. We explain how non-Gaussian features, e.g.~coherent structures and local spikes in the MHD magnetic field, are responsible for the heavy tail. Our results suggest that limits placed on ALPs using GRFs are robust., Comment: 18 pages, 9 figures. v2: major changes to match the published version. Extended appendix to include details on the statistical analysis

We show that in theories of axionlike particles (ALPs) coupled to electrons at tree-level, the one-loop effective coupling to photons is process dependent: the effective coupling relevant for decay processes, $g_{a\gamma}^{\text{(D)}}$, differs significantly from the coupling appearing in the phenomenologically important Primakoff process, $g_{a\gamma}^{\text{(P)}}$. We show that this has important implications for the physics of massive ALPs in hot and dense environments, such as supernovae. We derive, as a consequence, new limits on the ALP-electron coupling, $\hat{g}_{ae}$, from SN 1987A by accounting for all relevant production processes, including one-loop processes, and considering bounds from excess cooling as well as the absence of an associated gamma-ray burst from ALP decays. Our limits are among the strongest to date for ALP masses in the range $0.03 \, \text{MeV} \, < m_a< 240 \, \text{MeV}$. Moreover, we also show how cosmological bounds on the ALP-photon coupling translate into new, strong limits on $\hat{g}_{ae}$ at one loop. Our analysis emphasises that large hierarchies between ALP effective couplings are difficult to realise once quantum loops are taken into account., Comment: 36 pages, 10 figures. Changes in v2: gravitational redshift taken into account, small corrections. This is an author-created, un-copyedited version of an article published in JCAP. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it

Axion-like particles (ALPs) are a well-motivated extension to the standard model of particle physics, and X-ray observations of cluster-hosted AGN currently place the most stringent constraints on the ALP coupling to electromagnetism, $g_{a \gamma}$, for very light ALPs ($m_a\lesssim10^{-11}$ eV). We revisit limits obtained by Reynolds et al. (2020) using Chandra X-ray grating spectroscopy of NGC 1275, the central AGN in the Perseus cluster, examining the impact of the X-ray spectral model and magnetic field model. We also present a new publicly available code, ALPro, which we use to solve the ALP propagation problem. We discuss evidence for turbulent magnetic fields in Perseus and show that it can be important to resolve the magnetic field structure on scales below the coherence length. We re-analyse the NGC 1275 X-ray spectra using an improved data reduction and baseline spectral model. We find the limits are insensitive to whether a partially covering absorber is used in the fits. At low $m_a$ ($m_a\lesssim10^{-13}$ eV), we find marginally weaker limits on $g_{a \gamma}$ (by $0.1-0.3$ dex) with different magnetic field models, compared to Model B from Reynolds et al. (2020). A Gaussian random field (GRF) model designed to mimic $\sim50$ kpc scale coherent structures also results in only slightly weaker limits. We conclude that the existing Model B limits are robust assuming that $\beta_{\rm pl}\approx100$, and are insensitive to whether cell-based or GRF methods are used. However, astrophysical uncertainties regarding the strength and structure of cluster magnetic fields persist, motivating high sensitivity RM observations and tighter constraints on the radial profile of $\beta_{\rm pl}$., Comment: 17 pages, 9 figures, 2 tables. Accepted for publication in The Astrophysical Journal. ALPro code available at https://github.com/jhmatthews/alpro with accompanying documentation, as well as notebooks and data for reproducing the main results

Several laboratory experiments have published limits on axionlike particles (ALPs) with feeble couplings to electrons and masses in the keV-MeV range, under the assumption that such ALPs comprise the dark matter. We note that ALPs decay radiatively into photons, and show that for a large subset of the parameter space ostensibly probed by these experiments, the lifetime of the ALPs is shorter than the age of the universe. Such ALPs cannot consistently make up the dark matter, which significantly affects the interpretation of published limits from GERDA, Edelweiss-III, SuperCDMS and Majorana. Moreover, constraints from gamma-ray and X-ray astronomy exclude an even wider range of the ALP-electron coupling, and supersede all current experimental limits on dark matter ALPs in the 6 keV to 1 MeV mass range. These conclusions are rather model-independent, and can only be avoided at the expense of significant fine-tuning in theories where the ALP has additional couplings to other particles., Comment: 4 pages, 2 figures. v2: references added, typos corrected (results unchanged)

Axion-Like Particles (ALPs) are predicted by several Beyond the Standard Model theories, in particular, string theory. In the presence of an external magnetic field perpendicular to the direction of propagation, ALPs can couple to photons. Therefore, if an X-ray source is viewed through a magnetised plasma, such as a luminous quasar in a galaxy cluster, we may expect spectral distortions that are well described by photon-ALP oscillations. We present a $571 \ \mathrm{ks}$ combined High and Low Energy Transmission Grating (HETG/LETG) Chandra observation of the powerful radio-quiet quasar H1821+643, hosted by a cool-core cluster at redshift $0.3$. The spectrum is well described by a double power-law continuum and broad$+$narrow iron line emission typical of type-1 Active Galactic Nuclei (AGN), with remaining spectral features $< 2.5\%$. Using a cell-based approach to describe the turbulent cluster magnetic field, we compare our spectrum with photon-ALP mixing curves for 500 field realisations assuming that the thermal-to-magnetic pressure ratio remains constant up to the virial radius. At $99.7\%$ credibility and taking $\beta = 100$, we exclude all couplings $g_\mathrm{a\gamma} > 6.3 \times 10^{-13} \ {\mathrm{GeV}}^{-1}$ for most ALP masses $< 10^{-12} \ \mathrm{eV}$. Our results are moderately more sensitive to constraining ALPs than the best previous result from Chandra observations of the Perseus cluster, albeit with a less constrained field model. We reflect on the promising future of ALP studies with bright AGN embedded in rich clusters, especially with the upcoming Athena mission., Comment: 14 pages, 7 figures. Accepted for publication in MNRAS on 24th November 2021. Comments welcome

We study the weak mixing of photons and relativistic axion-like particles (axions) in plasmas with background magnetic fields, ${\bf B}$. We show that, to leading order in the axion-photon coupling, the conversion probability, $P_{\gamma \to a}$, is given by the one-dimensional power spectrum of the magnetic field components perpendicular to the particle trajectory. Equivalently, we express $P_{\gamma \to a}$ as the Fourier transform of the magnetic field autocorrelation function, and establish a dictionary between properties of the real-space magnetic field and the energy-dependent conversion probability. For axions more massive than the plasma frequency, ($m_a>\omega_{\rm pl}$), we use this formalism to analytically solve the problem of perturbative axion-photon mixing in a general magnetic field. In the general case where $\omega_{\rm pl}/m_a$ varies arbitrarily along the trajectory, we show that a naive application of the standard formalism for 'resonant' conversion can give highly inaccurate results, and that a careful calculation generically gives non-resonant contributions at least as large as the resonant contribution. Furthermore, we demonstrate how techniques based on the Fast Fourier Transform provide a new, highly efficient numerical method for calculating axion-photon mixing. We briefly discuss magnetic field modelling in galaxy clusters in the light of our results and argue, in particular, that a recently proposed 'regular' model used for studying axion-photon mixing (specifically applied to the Perseus cluster) is inconsistent with observations. Our formalism suggest new methods to search for imprints of axions, and will be important for spectrographs with percent level sensitivity, which includes existing X-ray observations by Chandra as well as the upcoming Athena mission., Comment: 39 pages, 14 figures. v3: matches published version

Axion dark matter can resonantly convert to photons in the magnetosphere of neutron stars, possibly giving rise to radio signals observable on Earth. This method for the indirect detection of axion dark matter has recently received significant attention in the literature. The calculation of the radio signal is complicated by a number of effects; most importantly, the gravitational infall of the axions onto the neutron star accelerates them to semi-relativistic speed, and the neutron star magnetosphere is highly anisotropic. Both of these factors complicate the calculation of the conversion of axions to photons. In this work, we present the first fully three-dimensional calculation of the axion-photon conversion in highly magnetised anisotropic media. Depending on the axion trajectory, this calculation leads to orders-of-magnitude differences in the conversion compared to the simplified one-dimensional calculation used so far in the literature, altering the directionality of the produced photons. Our results will have important implications for the radio signal one would observe in a telescope., Comment: v3: Matches published version. 27 pages, 4 figures

Context. Solar Energetic Particles (SEPs) with energy in the GeV range can propagate to Earth from their acceleration region near the Sun and produce Ground Level Enhancements (GLEs). The traditional approach to interpreting and modelling GLE observations assumes particle propagation only parallel to the magnetic field lines of interplanetary space, i.e. it is spatially 1D. Recent measurements by PAMELA have characterised SEP properties at 1 AU for the ~100 MeV-1 GeV range at high spectral resolution. Aims. We model the transport of GLE-energy solar protons through the Interplanetary Magnetic Field (IMF) using a 3D approach, to assess the effect of the Heliospheric Current Sheet (HCS) and drifts associated to the gradient and curvature of the Parker spiral. The latter are influenced by the IMF polarity. We derive 1 AU observables and compare the simulation results with data from PAMELA. Methods. We use a 3D test particle model including a HCS. Monoenergetic populations are studied first to obtain a qualitative picture of propagation patterns and numbers of crossings of the 1 AU sphere. Simulations for power law injection are used to derive intensity profiles and fluence spectra at 1 AU. A simulation for a specific event, GLE 71, is used to compare with PAMELA data. Results. Spatial patterns of 1 AU crossings and the average number of crossings are strongly influenced by 3D effects, with significant differences between periods of A+ and A- polarities. The decay time constant of 1 AU intensity profiles varies depending on the polarity and position of the observer, and it is not a simple function of the mean free path as in 1D models. Energy dependent leakage from the injection flux tube is particularly important for GLE energy particles, in many cases resulting in a roll-over in the fluence spectrum., Comment: Accepted by A&A (revised version, 9 pages, 6 figures)

Inflation can be supported in very steep potentials if it is generated by rapidly turning fields, which can be natural in negatively curved field spaces. The curvature perturbation, $\zeta$, of these models undergoes an exponential, transient amplification around the time of horizon crossing, but can still be compatible with observations at the level of the power spectrum. However, a recent analysis (based on a proposed single-field effective theory with an imaginary speed of sound) found that the trispectrum and other higher-order, non-Gaussian correlators also undergo similar exponential enhancements. This arguably leads to `hyper-large' non-Gaussianities in stark conflict with observations, and even to the loss of perturbative control of the calculations. In this paper, we provide the first analytic solution of the growth of the perturbations in two-field rapid-turn models, and find it in good agreement with previous numerical and single-field EFT estimates. We also show that the nested structure of commutators of the in-in formalism has subtle and crucial consequences: accounting for these commutators, we show analytically that the naively leading-order piece (which indeed is exponentially large) cancels exactly in all relevant correlators. The remaining non-Gaussianities of these models are modest, and there is no problem with perturbative control from the exponential enhancement of $\zeta$. Thus, rapid-turn inflation with negatively curved field spaces remains a viable and interesting class of candidate theories of the early universe., Comment: 19 pages, 2 figures

Axions/axion-like particles (ALPs) are a well motivated extension of the Standard Model and are generic within String Theory. The X-ray transparency of the intracluster medium (ICM) in galaxy clusters is a powerful probe of light ALPs (with mass $<10^{-11}\,{\rm eV}$); as X-ray photons from an embedded or background source propagate through the magnetized ICM, they may undergo energy-dependent quantum mechanical conversion into ALPs (and vice versa), imprinting distortions on the X-ray spectrum. We present Chandra data for the active galactic nucleus NGC1275 at the center of the Perseus cluster. Employing a 490ks High-Energy Transmission Gratings (HETG) exposure, we obtain a high-quality 1-9keV spectrum free from photon pileup and ICM contamination. Apart from iron-band features, the spectrum is described by a power-law continuum, with any spectral distortions at the $<3\%$ level. We compute photon survival probabilities as a function of ALP mass $m_a$ and ALP-photon coupling constant $g_{a\gamma}$ for an ensemble of ICM magnetic field models, and then use the NGC1275 spectrum to constraint the $(m_a, g_{a\gamma})$-plane. Marginalizing over magnetic field realizations, the 99.7% credible region limits the ALP-photon coupling to $g_{a\gamma}<6-8\times 10^{-13}\, {\rm GeV}^{-1}$ (depending upon magnetic field model) for masses $m_a<1\times 10^{-12}\,{\rm eV}$. These are the most stringent limit to date on $g_{a\gamma}$ for these light ALPs, and have already reached the sensitivity limits of next-generation helioscopes and light-shining-through-wall experiments. We highlight the potential of these studies with the next-generation X-ray observatories Athena and Lynx, but note the critical importance of advances in relative calibration of these future X-ray spectrometers., Comment: Published in The Astrophysical Journal, 2020, vol 890, pp59. This posting corrects typos in equation (2) and the likelihood function just prior to equation (3). Typos were introduced into the draft at late stage and the analysis is correct

We review the physics potential of a next generation search for solar axions: the International Axion Observatory (IAXO). Endowed with a sensitivity to discover axion-like particles (ALPs) with a coupling to photons as small as $g_{a\gamma}\sim 10^{-12}$ GeV$^{-1}$, or to electrons $g_{ae}\sim$10$^{-13}$, IAXO has the potential to find the QCD axion in the 1 meV$\sim$1 eV mass range where it solves the strong CP problem, can account for the cold dark matter of the Universe and be responsible for the anomalous cooling observed in a number of stellar systems. At the same time, IAXO will have enough sensitivity to detect lower mass axions invoked to explain: 1) the origin of the anomalous "transparency" of the Universe to gamma-rays, 2) the observed soft X-ray excess from galaxy clusters or 3) some inflationary models. In addition, we review string theory axions with parameters accessible by IAXO and discuss their potential role in cosmology as Dark Matter and Dark Radiation as well as their connections to the above mentioned conundrums., Comment: Prepared for submission to JCAP; new version with very minor corrections after referee review

Virtual globes have progressed from little-known technology to broadly popular software in a mere few years. We investigated this phenomenon through a survey and discovered that, while virtual globes are en vogue, their use is restricted to a small set of tasks so simple that they do not involve any spatial thinking. Spatial thinking requires that users ask "what is where" and "why"; the most common virtual globe tasks only include the "what". Based on the results of this survey, we have developed a multi-touch virtual globe derived from an adapted virtual globe paradigm designed to widen the potential uses of the technology by helping its users to inquire about both the "what is where" and "why" of spatial distribution. We do not seek to provide users with full GIS (geographic information system) functionality, but rather we aim to facilitate the asking and answering of simple "why" questions about general topics that appeal to a wide virtual globe user base., Comment: Proceedings of the International Conference on Intelligent User Interfaces (IUI 2008)

Our current understanding of the Universe is established through the pristine measurements of structure in the cosmic microwave background (CMB) and the distribution and shapes of galaxies tracing the large scale structure (LSS) of the Universe. One key ingredient that underlies cosmological observables is that the field that sources the observed structure is assumed to be initially Gaussian with high precision. Nevertheless, a minimal deviation from Gaussianityis perhaps the most robust theoretical prediction of models that explain the observed Universe; itis necessarily present even in the simplest scenarios. In addition, most inflationary models produce far higher levels of non-Gaussianity. Since non-Gaussianity directly probes the dynamics in the early Universe, a detection would present a monumental discovery in cosmology, providing clues about physics at energy scales as high as the GUT scale., Comment: 5 pages + references; Submitted to the Astro2020 call for science white papers. This version: fixed author list

In negatively curved field spaces, inflation can be realised even in steep potentials. Hyperinflation invokes the `centrifugal force' of a field orbiting the hyperbolic plane to sustain inflation. We generalise hyperinflation by showing that it can be realised in models with any number of fields ($N_f\geq2$), and in broad classes of potentials that, in particular, don't need to be rotationally symmetric. For example, hyperinflation can follow a period of radial slow-roll inflation that undergoes geometric destabilisation, yet this inflationary phase is not identical to the recently proposed scenario of `side-tracked inflation'. We furthermore provide a detailed proof of the attractor mechanism of (the original and generalised) hyperinflation, and provide a novel set of characteristic, explicit models. We close by discussing the compatibility of hyperinflation with observations and the recently much discussed `swampland conjectures'. Observationally viable models can be realised that satisfy either the `de Sitter conjecture' ($V'/V\gtrsim 1$) or the `distance conjecture' ($\Delta \phi \lesssim 1$), but satisfying both simultaneously brings hyperinflation in some tension with successful reheating after inflation. However, hyperinflation can get much closer to satisfying all of these criteria than standard slow-roll inflation. Furthermore, while the original model is in stark tension with the weak gravity conjecture, generalisations can circumvent this issue., Comment: 26 pages, 3 figures

It has recently been conjectured that string theory does not admit de Sitter vacua, and that quintessence explains the current epoch of accelerated cosmic expansion. A proposed, key prediction of this scenario is time-varying couplings in the dark sector, induced by the evolving quintessence field. We note that cosmological models with varying couplings suffer from severe problems with quantum corrections, beyond those shared by all quintessence models. The vacuum energy depends on all masses and couplings of the theory, and even small variations of parameters can lead to overwhelmingly large corrections to the effective potential. We find that quintessence models with varying parameters can be realised in consistent quantum theories by either: 1) enforcing exceptional levels of fine-tuning; 2) realising some unknown mechanism that cancels all undesirable contributions to the effective potential with unprecedented accuracy; or 3) ensuring that the quintessence field couples exclusively to very light states, and does not backreact on heavy fields., Comment: 4p

Many applications of Gaussian random fields and Gaussian random processes are limited by the computational complexity of evaluating the probability density function, which involves inverting the relevant covariance matrix. In this work, we show how that problem can be completely circumvented for the local Taylor coefficients of a Gaussian random field with a Gaussian (or `square exponential') covariance function. Our results hold for any dimension of the field and to any order in the Taylor expansion. We present two applications. First, we show that this method can be used to explicitly generate non-trivial potential energy landscapes with many fields. This application is particularly useful when one is concerned with the field locally around special points (e.g.~maxima or minima), as we exemplify by the problem of cosmic `manyfield' inflation in the early universe. Second, we show that this method has applications in machine learning, and greatly simplifies the regression problem of determining the hyperparameters of the covariance function given a training data set consisting of local Taylor coefficients at single point. An accompanying Mathematica notebook is available at https://doi.org/10.17863/CAM.22859 ., Comment: 15 pages, 2 figures

We consider the robustness of small-field inflation in the presence of scalar field inhomogeneities. Previous numerical work has shown that if the scalar potential is flat only over a narrow interval, such as in commonly considered inflection-point models, even small-amplitude inhomogeneities present at the would-be onset of inflation at $\tau = \tau_i$ can disrupt the accelerated expansion. In this paper, we parametrise and evolve the inhomogeneities from an earlier time $\tau_{IC}$ at which the initial data were imprinted, and show that for a broad range of inflationary and pre-inflationary models, inflection-point inflation withstands initial inhomogeneities. We consider three classes of perturbative pre-inflationary solutions (corresponding to energetic domination by the scalar field kinetic term, a relativistic fluid, and isotropic negative curvature), and two classes of exact solutions to Einstein's equations with large inhomogeneities (corresponding to a stiff fluid with cylindrical symmetry, and anisotropic negative curvature). We derive a stability condition that depends on the Hubble scales $H(\tau_ i)$ and $H(\tau_{IC})$, and a few properties of the pre-inflationary cosmology. For initial data imprinted at the Planck scale, the absence of an inhomogeneous initial data problem for inflection-point inflation leads to a novel, lower limit on the tensor-to-scalar ratio., Comment: 31 pages, 4 figures. Changes in v2: minor clarifications, some calculations moved to an appendix. This is an author-created, un-copyedited version of an article published in JCAP. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it

Forecasting Solar Energetic Particle (SEP) fluence, as integrated over an SEP event, is an important element when estimating the effect of solar eruptions on humans and technology in space. Current real-time estimates are based on SEP measurements at a single location in space. However, the interplanetary magnetic field corotates with the Sun approximately 13$^\circ$ each day with respect to Earth, thus in 4 days a near-Earth spacecraft will have changed their connection about 60$^\circ$ from the original SEP source. We estimate the effect of the corotation on particle fluence using a simple particle transport model, and show that ignoring corotation can cause up to an order of magnitude error in fluence estimations, depending on the interplanetary particle transport conditions. We compare the model predictions with STEREO observations of SEP events., Comment: 3 pages, 3 figures. Accepted for publication in Proceedings of IAU Symposium 335: Space Weather of the Heliosphere: Processes and Forecasts

We construct models of inflation with many randomly interacting fields and use these to study the generation of cosmological observables. We model the potentials as multi-dimensional Gaussian random fields (GRFs) and identify powerful algebraic simplifications that, for the first time, make it possible to access the manyfield limit of inflation in GRF potentials. Focussing on small-field, slow-roll, approximate saddle-point inflation in potentials with structure on sub-Planckian scales, we construct explicit examples involving up to 100 fields and generate statistical ensembles comprising of 164,000 models involving 5 to 50 fields. For the subset of these that support at least sixty e-folds of inflation, we use the 'transport method' and $\delta N$ formalism to determine the predictions for cosmological observables at the end of inflation, including the power spectrum and the local non-Gaussianities of the primordial perturbations. We find three key results: i) Planck compatibility is not rare, but future experiments may rule out this class of models; ii) In the manyfield limit, the predictions from these models agree well with, but are sharper than, previous results derived using potentials constructed through non-equilibrium Random Matrix Theory; iii) Despite substantial multifield effects, non-Gaussianities are typically very small: $f_{\rm nl}^{\rm loc} \ll 1$. We conclude that many of the 'generic predictions' of single-field inflation can be emergent features of complex inflation models., Comment: 44+16 pages, 18 figures; v2: references added, slightly extended discussion on reheating, published version

We construct a new class of metastable de Sitter vacua of flux compactifications of type IIB string theory. These solutions provide a natural extension of the `Large Volume Scenario' anti-de Sitter vacua, and can analogously be realised at parametrically large volume and weak string coupling, using standard ${\cal N}=1$ supergravity. For these new vacua, a positive vacuum energy is achieved from the inclusion of a small amount of flux-induced supersymmetry breaking in the complex structure and axio-dilaton sector, and no additional `uplift' contribution (e.g.~from anti-branes) is required. We show that the approximate no-scale structure of the effective theory strongly influences the spectrum of the stabilised moduli: one complex structure modulus remains significantly lighter than the supersymmetry breaking scale, and metastability requires only modest amounts of tuning. After discussing these general results, we provide a recipe for constructing de Sitter vacua on a given compactification manifold, and give an explicit example of a de Sitter vacuum for the compactification on the Calabi-Yau orientifold realised in $\mathbb{CP}^4_{11169}$. Finally, we note that these solutions have intriguing implications for phenomenology, predicting no superpartners in the spectrum below $\sim$50 TeV, and no WIMP dark matter., Comment: 34 pages

We study inflation in models with many interacting fields subject to randomly generated scalar potentials. We use methods from non-equilibrium random matrix theory to construct the potentials and an adaption of the 'transport method' to evolve the two-point correlators during inflation. This construction allows, for the first time, for an explicit study of models with up to 100 interacting fields supporting a period of 'approximately saddle-point' inflation. We determine the statistical predictions for observables by generating over 30,000 models with 2-100 fields supporting at least 60 efolds of inflation. These studies lead us to seven lessons: i) Manyfield inflation is not single-field inflation, ii) The larger the number of fields, the simpler and sharper the predictions, iii) Planck compatibility is not rare, but future experiments may rule out this class of models, iv) The smoother the potentials, the sharper the predictions, v) Hyperparameters can transition from stiff to sloppy, vi) Despite tachyons, isocurvature can decay, vii) Eigenvalue repulsion drives the predictions. We conclude that many of the 'generic predictions' of single-field inflation can be emergent features of complex inflation models., Comment: 53 pages, 19 figures. Minor edits to v1

Axion-like particles (ALPs) and photons can quantum mechanically interconvert when propagating through magnetic fields, and ALP-photon conversion may induce oscillatory features in the spectra of astrophysical sources. We use deep (370 ks), short frame time Chandra observations of the bright nucleus at the centre of the radio galaxy M87 in the Virgo cluster to search for signatures of light ALPs. The absence of substantial irregularities in the X-ray power-law spectrum leads to a new upper limit on the photon-ALP coupling, $g_{a\gamma}$: using a conservative model of the cluster magnetic field consistent with Faraday rotation measurements from M87 and M84, we find $g_{a \gamma} < 2.6\times10^{-12}$ GeV$^{-1}$ at 95% confidence level for ALP masses $m_a \leq 10^{-13}$ eV. Other consistent magnetic field models lead to stronger limits of $g_{a \gamma} \lesssim 1.1$--$1.5 \times 10^{-12}$ GeV$^{-1}$. These bounds are all stronger than the limit inferred from the absence of a gamma-ray burst from SN1987A, and rule out a substantial fraction of the parameter space accessible to future experiments such as ALPS-II and IAXO., Comment: 34 pages including references, 13 figures. Published version, additional figures added

The event-averaged charge state of heavy ion Solar Energetic Particles (SEPs), measured at 1 AU from the Sun, typically increases with the ions' kinetic energy. The origin of this behaviour has been ascribed to processes taking place within the acceleration region. In this paper we study the propagation through interplanetary space of SEP Fe ions, injected near the Sun with a variety of charge states that are uniformly distributed in energy, by means of a 3D test particle model. In our simulations, due to gradient and curvature drifts associated with the Parker spiral magnetic field, ions of different charge propagate with very different efficiencies to an observer that is not magnetically well connected to the source region. As a result we find that, for many observer locations, the 1 AU event-averaged charge state , as obtained from our model, displays an increase with particle energy E, in qualitative agreement with spacecraft observations. We conclude that drift-associated propagation is a possible explanation for the observed distribution of versus E in SEP events, and that the distribution measured in interplanetary space cannot be taken to represent that at injection., Comment: Submitted to ApJ, 2016

Warped string compactifications are central to many attempts to stabilize moduli and connect string theory with cosmology and particle phenomenology. We present a first-principles derivation of the low-energy 4D effective theory from dimensional reduction of a D3-brane in a warped Calabi-Yau compactification of type IIB string theory with imaginary self-dual 3-form flux, including effects of D3-brane motion beyond the probe approximation, and find the metric on the moduli space of brane positions, the universal volume modulus, and axions descending from the 4-form potential. As D3-branes may be considered as carrying either electric or magnetic charges for the self-dual 5-form field strength, we present calculations in both duality frames. Our results are consistent with, but extend significantly, earlier results on the low-energy effective theory arising from D3-branes in string compactifications., Comment: 37 pp + appendices; v2. typographical fixes, clarified section 4.2.3; v3. minor edits, added clarifications

Context. The intensity profiles of iron and oxygen in Solar Energetic Particle (SEP) events often display differences that result in a decreasing Fe/O ratio over time. The physical mechanisms behind this behaviour are not fully understood, but these observational signatures provide important tests of physical modelling efforts. Aims. In this paper we study the propagation of iron and oxygen SEP ions using a 3D model of propagation which includes the effect of guiding centre drift in a Parker spiral magnetic field. We derive time intensity profiles for a variety of observer locations and study the temporal evolution of the Fe/O ratio. Methods. We use a 3D full orbit test particle model which includes scattering. The configuration of the interplanetary magnetic field is a unipolar Parker spiral. Particles are released instantaneously from a compact region at 2 solar radii and allowed to propagate in 3D. Results. Both Fe and O experience significant transport across the magnetic field due to gradient and curvature drifts. We find that Fe ions drift more than O ions due to their larger mass-to-charge ratio, so that an observer that is not magnetically well connected to the source region will observe Fe arriving before O, for particles within the same range in energy per nucleon. As a result, for the majority of observer locations, the Fe/O ratio displays a decrease in time. Conclusions. We conclude that propagation effects associated with drifts produce a decay over time of the Fe/O ratio, qualitatively reproducing that observed in SEP event profiles., Comment: Submitted to A&A, revised version (6 pages, 6 figures)

Future cosmological surveys will probe the expansion history of the universe and constrain phenomenological models of dark energy. Such models do not address the fine-tuning problem of the vacuum energy, i.e. the cosmological constant problem (c.c.p.), but can make it spectacularly worse. We show that this is the case for 'interacting dark energy' models in which the masses of the dark matter states depend on the dark energy sector. If realised in nature, these models have far-reaching implications for proposed solutions to the c.c.p. that require the number of vacua to exceed the fine-tuning of the vacuum energy density. We show that current estimates of the number of flux vacua in string theory, $N_{\rm vac} \sim {\cal O}(10^{272,000})$, is far too small to realise certain simple models of interacting dark energy \emph{and} solve the cosmological constant problem anthropically. These models admit distinctive observational signatures that can be targeted by future gamma-ray observatories, hence making it possible to observationally rule out the anthropic solution to the cosmological constant problem in theories with a finite number of vacua., Comment: v2: 6 pages, 2 figures; extended discussion of observational prospects (conclusions unchanged); accepted to PRL

We construct ensembles of random scalar potentials for $N_f$ interacting scalar fields using non-equilibrium random matrix theory, and use these to study the generation of observables during small-field inflation. For $N_f={\cal O}({\rm few})$, these heavily featured scalar potentials give rise to power spectra that are highly non-linear, at odds with observations. For $N_f\gg 1$, the superhorizon evolution of the perturbations is generically substantial, yet the power spectra simplify considerably and become more predictive, with most realisations being well approximated by a linear power spectrum. This provides proof of principle that complex inflationary physics can give rise to simple emergent power spectra. We explain how these results can be understood in terms of large $N_f$ universality of random matrix theory., Comment: 5 pages, 2 figures, minor comments added to match PRL version

We consider flux compactifications of type IIB string theory and F-theory in which the respective superpotentials at large complex structure are dominated by cubic or quartic terms in the complex structure moduli. In this limit, the low-energy effective theory for the complex structure and axio-dilaton sector exhibits universal properties that are insensitive to the details of the compactification manifold or the flux configuration. We show that there are no vacua in this region and the spectrum of the Hessian matrix is highly peaked and consists only of three distinct eigenvalues ($0$, $2m_{3/2}^2$ and $8m_{3/2}^2$), independently of the number of moduli. We briefly comment on how the inclusion of K\"ahler moduli affect these findings. Our results generalise those of Brodie & Marsh [1], in which these universal properties were found in a subspace of the large complex structure limit of type IIB compactifications., Comment: 19 pages, published version

Axion-like particles (ALPs) and photons inter-convert in the presence of a magnetic field. At keV energies in the environment of galaxy clusters, the conversion probability can become unsuppressed for light ALPs. Conversion of thermal X-ray photons into ALPs can introduce a step-like feature into the cluster thermal bremsstrahlung spectrum, and we argue that existing X-ray data on galaxy clusters should be sufficient to extend bounds on ALPs in the low-mass region $m_a \lesssim 1 \times 10^{-12}\,{\rm eV}$ down to $M \sim 7\times 10^{11}\, {\rm GeV}$, and that for $10^{11}\, {\rm GeV} < M \lesssim 10^{12}$ GeV light ALPs give rise to interesting and unique observational signatures that may be probed by existing and upcoming X-ray (and potentially X-ray polarisation) observations of galaxy clusters., Comment: 5 pages

We compute the spectra of the Hessian matrix, ${\cal H}$, and the matrix ${\cal M}$ that governs the critical point equation of the low-energy effective supergravity, as a function of the complex structure and axio-dilaton moduli space in type IIB flux compactifications at large complex structure. We find both spectra analytically in an $h^{1,2}_-+3$ real-dimensional subspace of the moduli space, and show that they exhibit a universal structure with highly degenerate eigenvalues, independently of the choice of flux, the details of the compactification geometry, and the number of complex structure moduli. In this subspace, the spectrum of the Hessian matrix contains no tachyons, but there are also no critical points. We show numerically that the spectra of ${\cal H}$ and ${\cal M}$ remain highly peaked over a large fraction of the sampled moduli space of explicit Calabi-Yau compactifications with 2 to 5 complex structure moduli. In these models, the scale of the supersymmetric contribution to the scalar masses is strongly linearly correlated with the value of the superpotential over almost the entire moduli space, with particularly strong correlations arising for $g_s < 1$. We contrast these results with the expectations from the much-used continuous flux approximation, and comment on the applicability of Random Matrix Theory to the statistical modelling of the string theory landscape., Comment: 42 pages, 9 figures

Context. Current solar energetic particle (SEP) propagation models describe the effects of interplanetary plasma turbulence on SEPs as diffusion, using a Fokker-Planck (FP) equation. However, FP models cannot explain the observed fast access of SEPs across the average magnetic field to regions that are widely separated in longitude within the heliosphere without using unrealistically strong cross-field diffusion. Aims. We study whether the recently suggested early non-diffusive phase of SEP propagation can explain the wide SEP events with realistic particle transport parameters. Methods. We used a novel model that accounts for the SEP propagation along field lines that meander as a result of plasma turbulence. Such a non-diffusive propagation mode has been shown to dominate the SEP cross-field propagation early in the SEP event history. We compare the new model to the traditional approach, and to SEP observations. Results. Using the new model, we reproduce the observed longitudinal extent of SEP peak fluxes that are characterised by a Gaussian profile with $\sigma=30-50^\circ$, while current diffusion theory can only explain extents of 11$^\circ$ with realistic diffusion coefficients. Our model also reproduces the timing of SEP arrival at distant longitudes, which cannot be explained using the diffusion model. Conclusions. The early onset of SEPs over a wide range of longitudes can be understood as a result of the effects of magnetic field-line random walk in the interplanetary medium and requires an SEP transport model that properly describes the non-diffusive early phase of SEP cross-field propagation., Comment: 9 pages, 6 figures. Accepted for publication in Astronomy & Astrophysics

We investigate the deceleration of Solar Energetic Particles (SEPs) during their propagation from the Sun through interplanetary space, in the presence of weak to strong scattering in a Parker spiral configuration, using relativistic full orbit test particle simulations. The calculations retain all three spatial variables describing particles' trajectories, allowing to model any transport across the magnetic field. Large energy change is shown to occur for protons, due to the combined effect of standard adiabatic deceleration and a significant contribution from particle drift in the direction opposite to that of the solar wind electric field. The latter drift-induced deceleration is found to have a stronger effect for SEP energies than for galactic cosmic rays. The kinetic energy of protons injected at 1 MeV is found to be reduced by between 35 and 90% after four days, and for protons injected at 100 MeV by between 20 and 55%. The overall degree of deceleration is a weak function of the scattering mean free path, showing that, although adiabatic deceleration plays a role, a large contribution is due to particle drift. Current SEP transport models are found to account for drift-induced deceleration in an approximate way and their accuracy will need to be assessed in future work.

We study ${\cal N}=1$ supergravity with $N>1$ chiral superfields in which one of the fields has a K\"ahler potential of exact no-scale type. Such systems admit de Sitter (dS) solutions in which supersymmetry is predominantly broken by the no-scale field, with only a small contribution to the breaking coming from the other fields. Metastable dS vacua of this type were recently shown to be achievable by the finetuning of an $N\times N$ sub-matrix of the Hessian matrix at the critical point. We show that perturbatively small deformations of the no-scale Minkowski vacuum into dS are only possible when the spectrum of the no-scale vacuum, besides the no-scale field, contain an additional massless mode. The no-scale structure allows for a decoupling of $N-2$ fields, and metastability can be achieved by the tuning of ${\cal O}(N^0)$ parameters. We illustrate this scenario in several examples, and derive a geometric condition for its realisation in type IIB string theory. Supergravities in which the complex structure moduli space is a symmetric space, such as the string theory inspired STU-models, are non-generic and realise a modified version of the scenario. For the STU-model with a single non-perturbative correction we present an explicit analytic family of dS solutions that includes examples with quantised fluxes satisfying the O3-plane tadpole condition., Comment: 30 pages

Here we report on the dynamics of the structural order parameter in a chromium film using synchrotron radiation in response to photo-induced ultra-fast excitations. Following transient optical excitations the effective lattice temperature of the film rises close to the N\'{e}el temperature and the charge density wave (CDW) amplitude is reduced. Although we expect the electronic charge ordering to vanish shortly after the excitation we observe that the CDW is never completely disrupted, which is revealed by its unmodified period at elevated temperatures. We attribute the persistence of the CDW to the long-lived periodic lattice displacement in chromium. The long-term evolution shows that the CDW revives to its initial strength within 1 ns, which appears to behave in accordance with the temperature dependence in equilibrium. This study highlights the fundamental role of the lattice distortion in charge ordered systems and its impact on the re-condensation dynamics of the charge ordered state in strongly correlated materials., Comment: 17 pages, 7 figures

A statistical analysis of the relationship between solar energetic particles (SEPs) and properties of solar flares and coronal mass ejections (CMEs) is presented. SEP events during solar cycle 23 are selected which are associated with solar flares originating on the visible hemisphere of the Sun and at least of magnitude M1. Taking into account all flares and CMEs that occurred during this period, the probability for the occurrence of an SEP event near Earth is determined. A strong rise of this probability is observed for increasing flare intensities, more western locations, larger CME speeds and halo CMEs. The correlations between the proton peak flux and these solar parameters are derived for a low (>10 MeV) and high (>60 MeV) energy range excluding any flux enhancement due to the passage of fast interplanetary shocks. The obtained correlation coefficients are: 0.55+-0.07 (0.63+-0.06) with flare intensity and 0.56+-0.08 (0.40+-0.09) with the CME speed for E>10 MeV (E>60 MeV). For both energy ranges, the correlations with flare longitude and CME width are very small or non-existent. Furthermore, the occurrence probabilities, correlation coefficients and mean peak fluxes are derived in multi-dimensional bins combining the aforementioned solar parameters. The correlation coefficients are also determined in different proton energy channels ranging from 5 to 200 MeV. The results show that the correlation between the proton peak flux and the CME speed decreases with energy, while the correlation with the flare intensity shows the opposite behavior. Furthermore, the correlation with the CME speed is stronger than the correlation with the flare intensity below 15 MeV and becomes weaker above 20 MeV. Excluding the flux enhancements due to interplanetary shocks, only a small but not very significant change is observed in the correlation between the peak flux below 7 MeV and the CME speed., Comment: v1: 38 pages, 24 figures, 17 tables, accepted for publication in Solar Physics; v2: editorial corrections

Motivated by the possibility of explaining the 3.5 keV line through dark matter decaying to axion-like particles that subsequently convert to photons, we study ALP-photon conversion for sightlines passing within 50 pc of the galactic centre. Conversion depends on the galactic centre magnetic field which is highly uncertain. For fields at low or mid-range of observational estimates (10--100 $\mu$G), no observable signal is possible. For fields at the high range of observational estimates (a pervasive poloidal mG field over the central 150 pc) it is possible to generate sufficient signal to explain recent observations of a 3.5 keV line in the galactic centre. In this scenario, the galactic centre line signal comes predominantly from the region with $z > 20$ pc, reconciling the results from the Chandra and XMM-Newton X-ray telescopes. The dark matter to ALP to photon scenario also naturally predicts the non-observation of the 3.5 keV line in stacked galaxy spectra. We further explore predictions for the line flux in galaxies and suggest a set of galaxies that is optimised for observing the 3.5 keV line in this model., Comment: 21 pages + Appendix, 5 figures

The capability to predict the parameters of an SEP event such as its onset, peak flux, and duration is critical to assessing any potential space weather impact. We present a new flexible modelling system simulating the propagation of Solar Energetic Particles (SEPs) from locations near the Sun to any given location in the heliosphere to forecast the SEP flux profiles. SPARX uses an innovative methodology, that allows implementation within an operational framework, to overcome the time constraints of test particle modelling of SEP profiles, allowing the production of near real time SEP nowcasts and forecasts, when paired with appropriate near real time triggers. SPARX has the capability to produce SEP forecasts within minutes of being triggered by observations of a solar eruptive event. The model is based on the test particle approach and is spatially 3D, thus allowing for the possibility of transport in the direction perpendicular to the magnetic field. The model naturally includes the effects of perpendicular propagation due to drifts and drift-induced deceleration. The modelling framework and the way in which parameters of relevance for Space Weather forecasting are obtained are described. The first results from the modelling system are presented. These results demonstrate that corotation and drift of SEP streams play an important role in shaping SEP flux profiles., Comment: Accepted in Space Weather

We show that the scattering of a general spin-0 sector of dark radiation off the pre-recombination thermal plasma results in undetectably small spectral distortions of the Cosmic Microwave Background., Comment: 21 pages + references, 5 figures. Published version