Cosmology seeks to characterise our Universe in terms of models based on well-understood and tested physics. Today we know our Universe with a precision that once would have been unthinkable. This book develops the entire mathematical, physical and statistical framework within which this has been achieved. It tells the story of how we arrive at our profound conclusions, starting from the early twentieth century and following developments up to the latest data analysis of big astronomical datasets. It provides an enlightening description of the mathematical, physical and statistical basis for understanding and interpreting the results of key space- and ground-based data. Subjects covered include general relativity, cosmological models, the inhomogeneous Universe, physics of the cosmic background radiation, and methods and results of data analysis. Extensive online supplementary notes, exercises, teaching materials, and exercises in Python make this the perfect companion for researchers, teachers and students in physics, mathematics, and astrophysics.

Precision Cosmology: The First Half Million Years

Large thermal fluctuations can destroy skyrmions - tiny speckles of rotated magnetization in two-dimensional films. Yet at low temperatures skyrmions are believed to be topologically stable, which makes them good candidates for data storage and processing. However, as the authors demonstrate here, even at zero temperature a nanoscale skyrmion is not protected against quantum decay. The theory is based on the imaginary-time dynamics of the skyrmion. Due to the fact that even the smallest skyrmion is still formed by many atomic spins, its quantum decay provides an example of a nanoscale Schr\"odinger's cat that one can study with modern measuring techniques.

Quantum collapse of a magnetic skyrmion

We study the impact of quenched disorder (random exchange couplings or site dilution) on easy-plane pyrochlore antiferromagnets. In the clean system, order by disorder selects a magnetically ordered state from a classically degenerate manifold. In the presence of randomness, however, different orders can be chosen locally depending on details of the disorder configuration. Using a combination of analytical considerations and classical Monte Carlo simulations, we argue that any long-range-ordered magnetic state is destroyed beyond a critical level of randomness where the system breaks into magnetic domains due to random exchange anisotropies, becoming, therefore, a glass of spin clusters, in accordance with the available experimental data. These random anisotropies originate from off-diagonal exchange couplings in the microscopic Hamiltonian, establishing their relevance to other magnets with strong spin-orbit coupling.

Cluster-Glass Phase in Pyrochlore XY Antiferromagnets with Quenched Disorder.

It is shown that energy and momentum are conserved during the runaway motion of a radiating charge and during free fall of a charge in a gravitational field. The runaway motion demonstrates the consistency of classical electrodynamics and the Lorentz–Abraham–Dirac equation. The important role of the Schott (acceleration) energy in this connection is made clear, and it is shown that the Schott energy is the part of the electromagnetic field energy that is proportional to (minus) the scalar product of the velocity and acceleration of a moving accelerated charged particle. The Schott energy is negative if the acceleration is in the same direction as the velocity and positive if it is opposite. During runaway motion, the Schott energy becomes more and more negative, and for a charged particle, it is localized at the particle. It is also shown that a proton and a neutron fall with the same acceleration in a uniform gravitational field, although the proton radiates and the neutron does not. The radiation energy...

The significance of the Schott energy for energy-momentum conservation of a radiating charge obeying the Lorentz–Abraham–Dirac equation

Motivated by the observation of biskyrmions in centrosymmetric magnetic films [X. Z. Yu et al., Nat. Commun. 5, 3198 (2014), W. Wang et al., Adv. Mater. 28, 6887 (2016)], we investigate analytically and numerically the stability of biskyrmions in films of finite thickness, taking into account the nearest-neighbor exchange interaction, perpendicular magnetic anisotropy (PMA), dipole-dipole interaction (DDI), and the discreteness of the atomic lattice. The biskyrmion is characterized by the topological charge $Q=2$, the spatial scale $\ensuremath{\lambda}$, and another independent length $d$ that can be interpreted as a separation of two $Q=1$ skyrmions inside a $Q=2$ topological defect in the background of uniform magnetization. We find that biskyrmions with $d$ of order $\ensuremath{\lambda}$ can be stabilized by the magnetic field within a certain range of the ratio of PMA to DDI in a film having a sufficient number of atomic layers ${N}_{z}$. The shape of biskyrmions has been obtained by the numerical minimization of the energy of interacting spins in a $1000\ifmmode\times\else\texttimes\fi{}1000\ifmmode\times\else\texttimes\fi{}{N}_{z}$ atomic lattice. It is close to the exact solution of the Belavin-Polyakov model when $d$ is below the width of the ferromagnetic domain wall. We compute the magnetic moment of a biskyrmion and discuss ways of creating biskyrmions in experiment.

Stabilty of biskyrmions in centrosymmetric magnetic films

The classical Abraham–Lorentz formula for the radiation reaction on a point charge suffers from two notorious defects: runaways and preacceleration. Recently, several authors have advocated as an alternative the Landau–Lifshitz formula, which has neither fault. The latter formula is often presented as an approximation to Abraham–Lorentz, raising the delicate question of how an approximation can be considered more accurate than the original. For a spherical shell of finite size, the equation for the radiation reaction is noncontroversial. We begin there, obtain the Abraham–Lorentz and Landau–Lifshitz expressions as limiting cases, and undertake some numerical studies to determine which is superior.

Abraham-Lorentz versus Landau-Lifshitz

We study the random field $xy$ spin model at $T=0$ numerically on lattices of up to $1000\ifmmode\times\else\texttimes\fi{}1000\ifmmode\times\else\texttimes\fi{}1000$ spins with emphasis on the weak random field. Our numerical method is physically equivalent to slow cooling in which the system is gradually losing energy and relaxing to an energy minimum. The system shows glass properties, the resulting spin states depending strongly on the initial conditions. Random initial condition for the spins leads to a vortex glass (VG) state with short-range spin-spin correlations defined by the average distance between vortex lines. Collinear and some other vortex-free initial conditions result in vortex-free ferromagnetic (F) states that have a lower energy. The energy difference between the F and VG states correlates with the vorticity of the VG state. The correlation functions in the F states agree with the Larkin-Imry-Ma theory at short distances. The hysteresis curves for a weak random field are dominated by topologically stable spin walls ruptured by vortex loops. We find no relaxation paths from the F, VG, or any other states to the hypothetical vortex-free state with zero magnetization.

Random field x y model in three dimensions

We consider an n-component fixed-length order parameter interacting with a weak random field in d=1, 2, 3 dimensions. Relaxation from the initially ordered state and spin-spin correlation functions are studied on lattices containing hundreds of millions of sites. At n ≤ d the presence of topological defects leads to strong metastability and glassy behavior, with the final state depending on the initial condition. At n=d+1, when topological structures are nonsingular, the system possesses a weak metastability. At n>d+1, when topological objects are absent, the final, lowest-energy state is independent of the initial condition. It is characterized by the exponential decay of correlations that agrees quantitatively with the theory based upon the Imry-Ma argument.

Random fields, topology, and the Imry-Ma argument.

Scaling of coercivity in a 3d random anisotropy model

We study vortex states in a 3d random-field xy model of up to one billion lattice spins at T = 0. Starting with random spin orientations, the sample freezes into the vortex-glass state with a stretched-exponential decay of spin correlations, having short correlation length and a low susceptibility, compared to vortex-free states. In a field opposite to the initial magnetization, peculiar topological objects —walls of spins still opposite to the field— emerge along the hysteresis curve. On increasing the field strength, the walls develop cracks bounded by vortex loops. The loops then grow in size and eat the walls away. Applications to magnets and superconductors are discussed.

T. Proctor

Papers

Abraham-Lorentz versus Landau-Lifshitz

Random field x y model in three dimensions

Random fields, topology, and the Imry-Ma argument.

Scaling of coercivity in a 3d random anisotropy model

The role of vortices in the three-dimensional random-field xy model