Non-trivial Spatial Topology Research Articles

Cosmic topology. Part IVa. Classification of manifolds using machine learning: a case study with small toroidal universes

Non-trivial spatial topology of the Universe may give rise to potentially measurable signatures in the cosmic microwave background. We explore different machine learning approaches to classify harmonic-space realizations of the microwave background in the test case of Euclidean E 1 topology (the 3-torus) with a cubic fundamental domain of a size scale significantly smaller than the diameter of the last scattering surface. This is the first step toward developing a machine learning approach to classification of cosmic topology and likelihood-free inference of topological parameters. Different machine learning approaches are capable of classifying the harmonic-space realizations with accuracy greater than 99% if the topology scale is half of the diameter of the last-scattering surface and orientation of the topology is known. For distinguishing random rotations of these sky realizations from realizations of the covering space, the extreme gradient boosting classifier algorithm performs best with an accuracy of 88%. Slightly lower accuracies of 83% to 87% are obtained with the random forest classifier along with one- and two-dimensional convolutional neural networks. The techniques presented here can also accurately classify non-rotated cubic E 1 topology realizations with a topology scale slightly larger than the diameter of the last-scattering surface, if enough training data are provided. While information compressing methods like most machine learning approaches cannot exceed the statistical power of a likelihood-based approach that captures all available information, they potentially offer a computationally cheaper alternative. A principle challenge appears to be accounting for arbitrary orientations of a given topology, although this is also a significant hurdle for likelihood-based approaches.

Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologies E 1–E 6, E 11, E 12, E 16, and E 18 that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys.

Dimensionally restricted causal set quantum gravity: examples in two and three dimensions

We study dimensionally restricted non-perturbative causal set quantum dynamics in 2 and 3 spacetime dimensions with non-trivial global spatial topology. The causal set sample space is generated from causal embeddings into spacetime lattices with global spatial topology S1 and T2 in 2 and 3 dimensions, respectively. The quantum gravity partition function over these sample spaces is studied using Markov chain Monte Carlo (MCMC) simulations after analytic continuation. In both 2 and 3 dimensions we find a phase transition that separates the dominance of the action from that of the entropy. The action dominated phase is characterised by ‘layered’ posets with a high degree of connectivity, while the causal sets in the entropy dominated phase are manifold-like. This phase transition is similar in character to that seen for the sample space of 2-orders, which are topologically trivial, hence suggesting that this is a generic feature of dimensionally restricted sample spaces. The simulations use a newly developed framework for causal set MCMC calculations. Ours is the first implementation of a causal set dynamics restricted to 3 dimensions.

Axion detection via topological Casimir effect

We propose a new table-top experimental configuration for the direct detection of dark matter axions with mass in the $(10^{-6} \rm eV - 10^{-2} \rm eV)$ range using non-perturbative effects in a system with non-trivial spatial topology. Different from most experimental setups found in literature on direct dark matter axion detection, which relies on $\dot{\theta}$ or $\vec{\nabla}\theta$, we found that our system is in principle sensitive to a static $\theta\geq 10^{-14}$ and can also be used to set limit on the fundamental constant $\theta_{\rm QED}$ which becomes the fundamental observable parameter of the Maxwell system if some conditions are met. Connection with Witten effect when the induced electric charge $e'$ is proportional to $\theta$ and the magnetic monopole becomes the dyon with non-vanishing $e'=-e \frac{\theta}{2\pi}$ is also discussed.

Quantum entanglement in topological phases on a torus

In this paper we study the effect of non-trivial spatial topology on quantum entanglement by examining the degenerate ground states of a topologically ordered system on torus. Using the string-net (fixed-point) wave-function, we propose a general formula of the reduced density matrix when the system is partitioned into two cylinders. The cylindrical topology of the subsystems makes a significant difference in regard to entanglement: a global quantum number for the many-body states comes into play, together with a decomposition matrix $M$ which describes how topological charges of the ground states decompose into boundary degrees of freedom. We obtain a general formula for entanglement entropy and generalize the concept of minimally entangled states to minimally entangled sectors. Concrete examples are demonstrated with data from both finite groups and modular tensor categories (i.e., Fibonacci, Ising, etc.), supported by numerical verification.

Limits of the circles-in-the-sky searches in the determination of cosmic topology of nearly flat universes

[Abridged] An observable signature of a detectable nontrivial spatial topology of the Universe is the circles-in-the-sky in the CMB sky. In the most general search, pairs of circles with deviation from antipodality $0^\circ \leq \theta \leq 169^\circ$ and radii $10^\circ \leq \lambda \leq 90^\circ$ were investigated, but no matching circles were found. Assuming this negative result, we examine the question as to whether there are nearly flat universes with compact topology that would give rise to circles whose observable parameters $\lambda$ and $\theta$ fall o outside the ranges covered by this search. We derive the expressions for the deviation from antipodality and for the radius of the circles associated to a pair elements ($\gamma\,$,$\gamma^{-1}$) of the holonomy group $\Gamma$ which define the spatial section of any positively curved universe with a nontrivial topology. We show that there is a critical position that maximizes the deviation from antipodality, and prove that no matter how nearly flat the Universe is, it can always have a nontrivial spatial topology that gives rise to circles whose deviation from antipodality $\theta$ is larger than $169^\circ$, and whose radii of the circles $\lambda$ are smaller than $10^\circ$ for some observers. This makes apparent that slightly positively curved universes with cosmological parameters within Planck bounds can be endowed with a nontrivial spatial topology with values of the parameters $\lambda$ and $\theta$ outside the ranges covered by the searches for circles carried out so far. Thus, these circles searches so far undertaken are not sufficient to exclude the possibility of a universe with a detectable nontrivial cosmic topology. We present concrete examples of such nearly flat universes, and discuss the implications of our results in view of unavoidable practical limits of the circles-in-the-sky method.

Geometric phases generated by the non-trivial spatial topology of static vector fields linearly coupled to a neutral spin-endowed particle: application to 171Yb atoms trapped in a 2D optical lattice

We have constructed the geometric phases emerging from the non-trivial topology of a space-dependent magnetic field B(r), interacting with the spin magnetic moment of a neutral particle. Our basic tool, adapted from a previous work on Berry’s phases, is the space-dependent unitary transformation , which leads to the identity, , at each point r. In the ‘rotated’ Hamiltonian , is replaced by the non-Abelian covariant derivative where can be written as A1(r)Sx + A2(r)Sy + A3(r)Sz. The Abelian differentials Ak(r)·dr are given in terms of the Euler angles defining the orientation of B(r). The non-Abelian field transforms as a Yang–Mills field; however, its vanishing ‘curvature’ reveals its purely geometric character. We have defined a perturbation scheme based upon the assumption that in the longitudinal field A3(r) dominates the transverse field A1, 2(r) contributions, evaluated to second order. The geometry embedded in both the vector field A3(r) and the geometric magnetic field is described by their associated Aharonov–Bohm phase. As an illustration we study the physics of cold 171Yb atoms dressed by overlaying two circularly polarized stationary waves with orthogonal directions, which form a 2D square optical lattice. The frequency is tuned midway between the two hyperfine levels of the (6s6p)3P1 states to protect the optical B(r) field generated by the lattice from the dressed atom instability. The geometric field B3(r) is computed analytically in terms of the Euler angles. The magnitude of the second-order corrections due to the transverse fields can be reduced to the per cent level by a choice of light intensity that keeps the dressed atom loss rate ⩽5 s−1. A second optical lattice can be designed to confine the atoms inside 2D domains where . We extend our analysis to the case of a triangular lattice.

What can the detection of a single pair of circles-in-the-sky tell us about the geometry and topology of the Universe?

In a Universe with a detectable nontrivial spatial topology, the last scattering surface contains pairs of matching circles with the same distribution of temperature fluctuations---the so-called circles-in-the-sky. Searches undertaken for nearly antipodal pairs of such circles in cosmic microwave background maps have so far been unsuccessful. Previously, we had shown that the negative outcome of such searches, if confirmed, should in principle be sufficient to exclude a detectable nontrivial spatial topology for most observers in very nearly flat ($0<\ensuremath{\mid}{\ensuremath{\Omega}}_{\mathrm{tot}}\ensuremath{-}1\ensuremath{\mid}\ensuremath{\lesssim}{10}^{\ensuremath{-}5}$) (curved) universes. More recently, however, we have shown that this picture is fundamentally changed if the universe turns out to be exactly flat. In this case, there are many potential pairs of circles with large deviations from antipodicity that have not yet been probed by existing searches. Here, we study under what conditions the detection of a single pair of circles-in-the-sky can be used to uniquely specify the topology and the geometry of the spatial section of the Universe. We show that from the detection of a single pair of matching circles one can infer whether the spatial geometry is flat or not, and if so we show how to determine the topology (apart from one case) of the Universe using this information. An important additional outcome of our results is that the dimensionality of the circles-in-the-sky parameter space that needs to be spanned in searches for matching pairs of circles is reduced from six to 5 degrees of freedom, with a significant reduction in the necessary computational time.

Circles-in-the-sky searches and observable cosmic topology in a flat universe

[Abridged] In a Universe with a detectable nontrivial spatial topology the last scattering surface contains pairs of matching circles with the same distribution of temperature fluctuations - the so-called circles-in-the-sky. Searches for nearly antipodal circles in maps of cosmic microwave background have so far been unsuccessful. This negative outcome along with recent theoretical results concerning the detectability of nearly flat compact topologies is sufficient to exclude a detectable nontrivial topology for most observers in very nearly flat positively and negatively curved Universes ($0<|\Omega_{tot}-1| \lesssim 10^{-5}$). Here we investigate the consequences of these searches for observable nontrivial topologies if the Universe turns out to be exactly flat ($\Omega_{tot}=1$). We demonstrate that in this case the conclusions deduced from such searches can be radically different. We show that for all multiply-connected orientable flat manifolds it is possible to directly study the action of the holonomies in order to obtain a general upper bound on the angle that characterizes the deviation from antipodicity of pairs of matching circles associated with the shortest closed geodesic. This bound is valid for all observers and all possible values of the compactification length parameters. We also show that in a flat Universe there are observers for whom the circles-in-the-sky searches already undertaken are insufficient to exclude the possibility of a detectable nontrivial spatial topology. It is remarkable how such small variations in the spatial curvature of the Universe, which are effectively indistinguishable geometrically, can have such a drastic effect on the detectability of cosmic topology.

CONSTRAINTS ON DARK ENERGY AND COSMIC TOPOLOGY

A non-trivial spatial topology of the Universe is a potentially observable attribute, which can be probed through the circles-in-the-sky for all locally homogeneous and isotropic universes with no assumptions on the cosmological parameters. We show how one can use a possible circles-in-the-sky detection of the spatial topology of globally homogeneous universes to set constraints on the dark energy equation of state parameters.

ON AXIAL AND PLANE-MIRROR INHOMOGENEITIES IN THE WMAP3 COSMIC MICROWAVE BACKGROUND MAPS

We study inhomogeneities in the distribution of the excursion sets in the Cosmic Microwave Background (CMB) temperature maps obtained by the three years survey of the Wilkinson Microwave Anisotropy Probe (WMAP). At temperature thresholds |T| = 90 μ K , the distributions of the excursion sets with over 200 pixels are concentrated in two regions, nearly at the antipodes, with galactic coordinates l = 94.7°, b = 34.4° and l = 279.8°, b = -29.2°. The centers of these two regions drift towards the equator when the temperature threshold is increased. The centers are located close to one of the vectors of ℓ = 3 multipole. The two patterns of the substructures in the distribution of the excursion sets are mirrored, with χ2 = 0.7–1.5. There is no obvious origin of this effect in the noise structure of WMAP, and there is no evidence for a dependence on the galactic cut. Would this effect be cosmological, it could be an indication of an anomalously large component of horizon-size density perturbations, independent of one of the spatial coordinates, and/or of a non-trivial slab-like spatial topology of the Universe.

Dodecahedral topology fails to explain quadrupole–octupole alignment

The cosmic microwave background (CMB) quadrupole and octupole, as well as being weaker than expected, align suspiciously well with each other. The non-trivial spatial topology can explain the weakness. Might it also explain the alignment? The answer in the case of the Poincaré dodecahedral space is, as one might expect, a resounding no.

Coexistence and shell structures of several superfluids in trapped three-component Fermi mixtures

We study the properties of a trapped interacting three component Fermi gas. We assume that one of the components can have a different mass from the other two. We calculate the different phases of the three component mixture and find a rich variety of different phases corresponding to different pairing channels, and simple ways of tuning the system from one phase to another. In particular, we predict coexistence of several different superfluids in the trap, forming a shell structure, and phase transitions from this mixture of superfluids to a single superfluid when the system parameters or temperature is varied. Such shell structures realize superfluids with a nontrivial spatial topology and leave clear observable signatures in the density profile of the gas.

A spin-statistics theorem for certain topological geons

We review the mechanism in quantum gravity whereby topological geons, particles made from non-trivial spatial topology, are endowed with non-trivial spin and statistics. In a theory without topology change there is no obstruction to `anomalous' spin-statistics pairings for geons. However, in a sum-over-histories formulation including topology change, we show that non-chiral Abelian geons do satisfy a spin-statistics correlation if they are described by a wavefunction which is given by a functional integral over metrics on a particular 4-manifold. This manifold describes a topology changing process which creates a pair of geons from .

Fitting Hyperbolic Universes to Cayon-Smoot Spots in COBE Maps

view Abstract Citations (15) References (23) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Fitting Hyperbolic Universes to Cayon-Smoot Spots in COBE Maps Fagundes, Helio V. Abstract On the possibility that the universe's matter density is low (Ω0 < 1), cosmologies can be considered with the metric of Friedmann's open universe but with closed hyperbolic manifolds as the physical three-space. These models have nontrivial spatial topology, with the property of producing multiple images of cosmic sources. Here a fit is attempted of 10 of these models to the physical cold and hot spots found by Cayón & Smoot in the COBE/DMR maps. These spots are interpreted as early, distant images of much nearer sources of inhomogeneity. The source for one of the cold spots is seen as the seed of a known supercluster. Publication: The Astrophysical Journal Pub Date: October 1996 DOI: 10.1086/177847 Bibcode: 1996ApJ...470...43F Keywords: COSMOLOGY: THEORY; COSMOLOGY: COSMIC MICROWAVE BACKGROUND full text sources ADS |

The Angular Power Spectrum of the Four-Year [ITAL]COBE[/ITAL] Data

The angular power spectrum Cℓ is extracted from the 4 yr COBE Differential Microwave Radiometer (DMR) data with a 20° Galactic cut, using the narrowest window functions possible. The average power in eight multipole bands is also computed and plotted together with a compilation of power spectrum measurements from other experiments. The COBE results are found to be consistent with an n = 1 power spectrum with the normalization Qrms, ps = 18 μK reported by the COBE DMR team. Certain nonstandard cosmologies, such as "small universe" models with nontrivial spatial topology, predict power spectra that are not smooth functions. Rather, they contain bumps and wiggles that may not have been detected by other data analysis techniques such as the Hauser-Peebles method, the band-power method, and orthogonalized spherical harmonics, since these methods all give broader window functions that can smear such features out beyond recognition. On the large angular scales probed by COBE, the universe thus appears to be kind to us, presenting a power spectrum that is a simple smooth function.

Minimal surfaces, spatial topology and singularities in space-time

Using an approach first advocated by Gannon (1975, 1976) and recent results of Meeks, Simon and Yau (1982) on the existence of compact minimal surfaces some new results are obtained relating non-trivial spatial topology to the occurrence of singularities in space-time. For example, it is shown that if V3 is a contracting body with mean-convex boundary homeomorphic to a two-sphere in a space-time M4 obeying appropriate curvature and causality assumptions, then either V3 is a three-cell or M4 is non-space-like geodesically incomplete.

The construction from initial data of spacetimes with nontrivial spatial and bundle topology

We show how to use the 3 + 1 construction program to build globally heperbolic spacetimes with topologically nontrivial Cauchy surfaces. Spacetimes in which the classical fields are sections of a nontrivial bundle are handled as well. In evolving the initial data in these spacetimes, one must work with an atlas of overlapping patches. Data must be transfered from patch to patch during the evolution, so the transition functions on patch intersections must be evolved as well. We describe how to do this. Often the evolution of the Cauchy data is considerably simplified by choosing the coordinate-shift field M and the gaugeshift field A ↓ to be patch dependent. We give examples of this phenomenon and show how to incorporate the patch dependence of M and A ↓ into a consistent evolution program for the spacetime.

The construction from initial data of spacetimes with nontrivial spatial and bundle topology: James A. Isenberg. Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

The construction from initial data of spacetimes with nontrivial spatial and bundle topology: James A. Isenberg. Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada