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1. The topology of general cosmological models*

   https://doi.org/10.1088/1361-6382/ac75e1  

   Galloway, Gregory J. ; Khuri, Marcus A. ; Woolgar, Eric ( August 2022 , Classical and Quantum Gravity)

   Is the Universe finite or infinite, and what shape does it have? These fundamental questions, of which relatively little is known, are typically studied within the context of the standard model of cosmology where the Universe is assumed to be homogeneous and isotropic. Here we address the above questions in highly general cosmological models, with the only assumption being that the average flow of matter is irrotational. Using techniques from differential geometry, specifically extensions of the Bonnet–Myers theorem, we derive a condition which implies a finite Universe and yields a bound for its diameter. Furthermore, under a weaker condition involving the interplay between curvature and diameter, together with the assumption that the Universe is finite (i.e. has closed spatial slices), we provide a concise list of possible topologies. Namely, the spatial sections then would be either the ring topologiesS¹×S²,$S^1\tilde{\times }{S}^2$,$S^1\times \mathbb{R}{\mathbb{P}}^2$,$\mathbb{R}{\mathbb{P}}^3\#\mathbb{R}{\mathbb{P}}^3$, or covered by the sphereS³or torusT³. In particular, under this condition the basic construction of connected sums would be ruled out (save for one), along with the plethora of topologies associated with negative curvature. These results are obtained frommore »consequences of the geometrization of three-manifolds, by applying a generalization of the almost splitting theorem together with a curvature formula of Ehlers and Ellis.

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2. Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

   https://doi.org/10.1088/1361-6544/ac50bb  

   Chen, Qinbo ; de la Llave, Rafael ( March 2022 , Nonlinearity)

   The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the followinga prioriunstable Hamiltonian system with a time-periodic perturbation${\mathcal{H}}_{\epsilon }(p,q,I,\phi ,t)=h(I)+\sum _{i=1}^n\pm \left(\frac{1}{2}{p}_i^2+V_i(q_i)\right)+\epsilon H_1(p,q,I,\phi ,t),$where$(p,q)\in {\mathbb{R}}^n\times {\mathbb{T}}^n$,$(I,\phi )\in {\mathbb{R}}^d\times {\mathbb{T}}^d$withn,d⩾ 1,V_iare Morse potentials, andɛis a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbationsH₁. Indeed, the set of admissibleH₁isC^ωdense andC³open (a fortiori,C^ωopen). Our perturbative technique for the genericity is valid in theC^ktopology for allk∈ [3, ∞) ∪ {∞,ω}.
3. Constant spacing in filament bundles

   https://doi.org/10.1088/1367-2630/ab1c2d  

   Atkinson, Daria W. ; Santangelo, Christian D. ; Grason, Gregory M. ( June 2019 , New Journal of Physics)

   Assemblies of one-dimensional filaments appear in a wide range of physical systems: from biopolymer bundles, columnar liquid crystals, and superconductor vortex arrays; to familiar macroscopic materials, like ropes, cables, and textiles. Interactions between the constituent filaments in such systems are most sensitive to thedistance of closest approachbetween the central curves which approximate their configuration, subjecting these distinct assemblies to common geometric constraints. In this paper, we consider two distinct notions of constant spacing in multi-filament packings in${\mathbb{R}}^3$:equidistance, where the distance of closest approach is constant along the length of filament pairs; andisometry, where the distances of closest approach between all neighboring filaments are constant and equal. We show that, although any smooth curve in${\mathbb{R}}^3$permits one dimensional families of collinear equidistant curves belonging to a ruled surface, there are only two families of tangent fields with mutually equidistant integral curves in${\mathbb{R}}^3$. The relative shapes and configurations of curves in these families are highly constrained: they must be either (isometric) developable domains, which can bend, but not twist; or (non-isometric) constant-pitch helical bundles, which can twist, but not bend. Thus, filament textures that are simultaneously bent and twisted, suchmore »as twisted toroids of condensed DNA plasmids or wire ropes, are doubly frustrated: twist frustrates constant neighbor spacing in the cross-section, while non-equidistance requires additional longitudinal variations of spacing along the filaments. To illustrate the consequences of the failure of equidistance, we compare spacing in three ‘almost equidistant’ ansatzes for twisted toroidal bundles and use our formulation of equidistance to construct upper bounds on the growth of longitudinal variations of spacing with bundle thickness.

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4. Liquid crystal director fields in three-dimensional non-Euclidean geometries

   https://doi.org/10.1088/1367-2630/abaf6c  

   Sadoc, Jean-François ; Mosseri, Rémy ; Selinger, Jonathan V. ( September 2020 , New Journal of Physics)

   This paper investigates nematic liquid crystals in three-dimensional curved space, and determines which director deformation modes are compatible with each possible type of non-Euclidean geometry. Previous work by Sethnaet alshowed that double twist is frustrated in flat space${\mathbb{R}}^3$, but can fit perfectly in the hypersphere${\mathbb{S}}^3$. Here, we extend that work to all four deformation modes (splay, twist, bend, and biaxial splay) and all eight Thurston geometries. Each pure mode of director deformation can fill space perfectly, for at least one type of geometry. This analysis shows the ideal structure of each deformation mode in curved space, which is frustrated by the requirements of flat space.
5. (Nearly) Model-independent Constraints on the Neutral Hydrogen Fraction in the Intergalactic Medium at z ∼ 5–7 Using Dark Pixel Fractions in Lyα and Lyβ Forests

   https://doi.org/10.3847/1538-4357/aca678  

   Jin, Xiangyu ; Yang, Jinyi ; Fan, Xiaohui ; Wang, Feige ; Bañados, Eduardo ; Bian, Fuyan ; Davies, Frederick B. ; Eilers, Anna-Christina ; Farina, Emanuele Paolo ; Hennawi, Joseph F. ; et al ( January 2023 , The Astrophysical Journal)

   Cosmic reionization was the last major phase transition of hydrogen from neutral to highly ionized in the intergalactic medium (IGM). Current observations show that the IGM is significantly neutral atz> 7 and largely ionized byz∼ 5.5. However, most methods to measure the IGM neutral fraction are highly model dependent and are limited to when the volume-averaged neutral fraction of the IGM is either relatively low (${\overline{x}}_{\mathrm{H}\mathrm{I}}\lesssim {10}^{-3}$) or close to unity (${\overline{x}}_{\mathrm{H}\mathrm{I}}\sim 1$). In particular, the neutral fraction evolution of the IGM at the critical redshift range ofz= 6–7 is poorly constrained. We present new constraints on${\overline{x}}_{\mathrm{H}\mathrm{I}}$atz∼ 5.1–6.8 by analyzing deep optical spectra of 53 quasars at 5.73 <z< 7.09. We derive model-independent upper limits on the neutral hydrogen fraction based on the fraction of “dark” pixels identified in the Lyαand Lyβforests, without any assumptions on the IGM model or the intrinsic shape of the quasar continuum. They are the first model-independent constraints on the IGM neutral hydrogen fraction atz∼ 6.2–6.8 using quasar absorption measurements. Our results give upper limits ofmathvariant='normal'>HI(z=6.3)<0.79±0.04(1σ),${\overline{x}}_{\mathrm{H}\mathrm{I}}(z=6.5)<0.87\pm 0.03$(1σ), and${\overline{x}}_{\mathrm{H}\mathrm{I}}(z=6.7)<{0.94}_{-0.09}^{+0.06}$(1σ). The dark pixel fractions atz> 6.1 are consistent with the redshift evolution of the neutral fraction of the IGM derived from Planck 2018.

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