Tag Archives: Mathematical Physics

Spectrum of the linearized Vlasov–Poisson equation around steady states from galactic dynamics [CL]

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http://arxiv.org/abs/2305.05749


We study the linearized Vlasov-Poisson equation in the gravitational case around steady states that are decreasing and continuous functions of the energy. We identify the absolutely continuous spectrum and give criteria for the existence of oscillating modes and estimate their number. Our method allows us to take into account an attractive external potential.

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M. Moreno, P. Rioseco and H. Bosch
Thu, 11 May 23
5/55

Comments: 17 pages, 2 figures

Horizon fluxes of binary black holes in eccentric orbits [CL]

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http://arxiv.org/abs/2305.03771


I compute the rate of change of mass and angular momentum of a black hole, namely tidal heating, in an eccentric orbit. The change is caused due to the tidal field of the orbiting companion. I compute the result for both the spinning and non-spinning black holes in the leading order of the mean motion, namely $\xi$. I demonstrate that the rates get enhanced significantly for nonzero eccentricity. Since eccentricity in a binary evolves with time I also express the results in terms of an initial eccentricity and azimuthal frequency $\xi_{\phi}$. In the process, I developed a prescription that can be used to compute all physical quantities in a series expansion of initial eccentricity, $e_0$. This result was only known in the leading order while ignoring the contribution of the spin on the eccentricity evolution. Although the eccentricity evolution result still ignores the spin effect in the current work, the prescription can be used to compute higher-order corrections of initial eccentricity post-leading order. Using this result I computed the rate of change of mass and angular momentum of a black hole in terms of initial eccentricity and azimuthal frequency up to $\mathcal{O}(e_0^2)$.

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S. Datta
Tue, 9 May 23
21/88

Comments: N/A

Complex evaluation of angular power spectra: Going beyond the Limber approximation [CEA]

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http://arxiv.org/abs/2304.13064


Angular power spectra are central to the study of our Universe. In this paper, I develop a new method for the numeric evaluation and analytic estimation of the angular cross-power spectrum of two random fields using complex analysis and Picard- Lefschetz theory. The proposed continuous deformation of the integration domain resums the highly oscillatory integral into a convex integral whose integrand decays exponentially. This deformed integral can be quickly evaluated with conventional integration techniques. These methods can be used to quickly evaluate and estimate the angular power spectrum from the three-dimensional power spectrum for all angles (or multipole moments). This method is especially useful for narrow redshift bins, or samples with small redshift overlap, for which the Limber approximation has a large error.

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J. Feldbrugge
Thu, 27 Apr 23
57/78

Comments: N/A

Trajectories of astroparticles in pseudo-Finsler spacetime with the most general modified dispersion [CL]

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http://arxiv.org/abs/2304.08676


Finsler geometry is a natural and fundamental generalization of Riemann geometry, and is a tool to research Lorentz invariance violation. We find the connection between the most general modified dispersion relation and a pseudo-Finsler structure, and then we calculate the arrival time delay of astroparticles with different modified dispersion relations in the framework of Finsler geometry. The result suggests that the time delay is irrelevant with the exact form of the modified dispersion relation. If the modified term becomes 0 when $E=p$, there is no arrival time difference, otherwise the time delays only depend on the Lorentz violation scale and the order at which the Lorentz invariance breaks.

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J. Zhu and B. Ma
Wed, 19 Apr 23
53/58

Comments: 9 pages, no figure, version for journal publication

Relegation-free closed-form perturbation theory and the domain of secular motions in the Restricted 3-Body Problem [CL]

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http://arxiv.org/abs/2301.03070


We propose a closed-form (i.e. without expansion in the orbital eccentricities) scheme for computations in perturbation theory in the restricted three-body problem (R3BP) when the massless particle is in an orbit exterior to the one of the primary perturber. Starting with a multipole expansion of the barycentric (Jacobi-reduced) Hamiltonian, we carry out a sequence of normalizations in Delaunay variables by Lie series, leading to a secular Hamiltonian model without use of relegation. To this end, we introduce a book-keeping analogous to the one proposed in Cavallari and Efthymiopoulos (2022) for test particle orbits interior to the one of the primary perturber, but here adapted, instead, to the case of exterior orbits. We give numerical examples of the performance of the method in both the planar circular and the spatial elliptic restricted three-body problem, for parameters pertinent to the Sun-Jupiter system. In particular, we demonstrate the method’s accuracy in terms of reproducibility of the orbital elements’ variations far from mean-motion resonances. As a basic outcome of the method, we show how, using as criterion the size of the series’ remainder, we reach to obtain an accurate semi-analytical estimate of the boundary (in the space of orbital elements) where the secular Hamiltonian model arrived at after eliminating the particle’s fast degree of freedom provides a valid approximation of the true dynamics.

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M. Rossi and C. Efthymiopoulos
Tue, 10 Jan 23
87/93

Comments: N/A

Dynamics of interacting monomial scalar field potentials and perfect fluids [CL]

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http://arxiv.org/abs/2212.02942


Motivated by cosmological models of the early universe we analyse the dynamics of the Einstein equations with a minimally coupled scalar field with monomial potentials $V(\phi)=\frac{(\lambda\phi)^{2n}}{2n}$, $\lambda>0$, $n\in\mathbb{N}$, interacting with a perfect fluid with linear equation of state $p_\mathrm{pf}=(\gamma_\mathrm{pf}-1)\rho_\mathrm{pf}$, $\gamma_\mathrm{pf}\in(0,2)$, in flat Robertson-Walker spacetimes. The interaction is a friction-like term of the form $\Gamma(\phi)=\mu \phi^{2p}$, $\mu>0$, $p\in\mathbb{N}\cup{0}$. The analysis relies on the introduction of a new regular 3-dimensional dynamical systems’ formulation of the Einstein equations on a compact state space, and the use of dynamical systems’ tools such as quasi-homogeneous blow-ups and averaging methods involving a time-dependent perturbation parameter. We find a bifurcation at $p=n/2$ due to the influence of the interaction term. In general, this term has more impact on the future (past) asymptotics for $p<n/2$ ($p>n/2$). For $p<n/2$ we find a complexity of possible future attractors, which depends on whether $p=(n-1)/2$ or $p<(n-1)/2$. In the first case the future dynamics is governed by Li\’enard systems. On the other hand when $p=(n-2)/2$ the generic future attractor consists of new solutions previously unknown in the literature which can drive future acceleration whereas the case $p<(n-2)/2$ has a generic future attractor de-Sitter solution. For $p=n/2$ the future asymptotics can be either fluid dominated or have an oscillatory behaviour where neither the fluid nor the scalar field dominates. For $p>n/2$ the future asymptotics is similar to the case with no interaction. Finally, we show that irrespective of the parameters, an inflationary quasi-de-Sitter solution always exists towards the past, and therefore the cases with $p\leq(n-2)/2$ may provide new cosmological models of quintessential inflation.

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A. Alho, V. Bessa and F. Mena
Wed, 7 Dec 22
71/74

Comments: 63 pages, 54 figures

Fully nonlinear Jeans instabilities for expanding Newtonian universes under homogeneous and isotropic perturbations [CL]

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http://arxiv.org/abs/2210.04657


Based on mathematically rigorous analysis of nonlinear differential equations studied in our companion article [1], we construct a model which describes the \textit{nonlinear} gravitational instability on a local portion of the universe characterized by the expanding Newtonian universe. In this portion, the perturbations are homogeneous and isotropic. This result, to some extent, can be viewed as a nonlinear version of the Jeans instability. The growth rate of the relative density due to the nonlinear effects is much faster (at least $\sim \exp(t^{\frac{2}{3}})$ or blowup at a finite time according to the data) than the one predicted by the classical linear version of the Jeans instability ($\sim t^{\frac{2}{3}}$), and it leads to a better, or potentially substantial impacts on, understanding of the formation of the nonlinear structures in the universe and stellar systems. This article associated with [1] provides a new window into the rigorously mathematical and robust method, instead of the most used approximations and numerical calculations, of the fully nonlinear analysis of the Jeans instability for general cases.

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C. Liu
Tue, 11 Oct 22
53/92

Comments: 15 pages

On the possible implications of Dark Matter in the rings of Saturn: a conjecture [CL]

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http://arxiv.org/abs/2210.01446


In this article we discuss some consequences of the well-known proposition of Fritz Zwicky [1], published in the nineteen thirties, that Dark Matter `mimics’ the inertia-gravitational behaviour of usual matter. In particular, we consider some special dynamical regions such as those of the Ring Systems of the gaseous giants at the edge of the Planetary System. This article is a continuation of an earlier paper [2], where it was shown that gravitationally interacting particles may remain near the Lagrange Points L4 and L5 for many thousands of years. This provides enough time for the Dark Matter, if present there, to interact with the usual matter. We discuss also a number of questions related to places which might be considered singular in the mathematical sense.

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A. Ciulli and S. Ciulli
Wed, 5 Oct 22
68/73

Comments: 14 pages, 9 figures

Regularization of the Hill four-body problem with oblate bodies [CL]

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http://arxiv.org/abs/2209.13625


We consider the Hill four-body problem where three oblate, massive bodies form a relative equilibrium triangular configuration, and the fourth, infinitesimal body orbits in a neighborhood of the smallest of the three massive bodies. We regularize collisions between the infinitesimal body and the smallest massive body, via McGehee coordinate transformation. We describe the corresponding collision manifold and show that it undergoes a bifurcation when the oblateness coefficient of the small massive body passes through the zero value.

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E. Belbruno, M. Gidea and W. Lam
Thu, 29 Sep 22
40/70

Comments: N/A

3-Body Problems, Hidden Constants, Trojans and WIMPs [CL]

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http://arxiv.org/abs/2209.10600


This work includes two new results – principally two new constants of motion for the linearised restricted 3-body problem (e.g. for the Trojan asteroids) and an important isosceles triangle generalisation of Lagrange’s equilateral triangle solution of the restricted case leading to hidden constants for Hildans as well as Trojans. Both of these results are classical, but we also have included new results on Newtonian quantum gravity emanating from the asymptotics relevant for WIMPish particles, explaining the origin of systems like that of the Trojans. The latter result uses a generalisation of our semi-classical mechanics for Schr\”odinger equations involving vector as well as scalar potentials, presented here for the first time, thereby providing an acid test of our ideas in predicting the quantum curvature and torsion of WIMPish trajectories for our astronomical elliptic states. The combined effect is to give a new celestial mechanics for WIMPs in gravitational systems as well as new results for classical problems. As we shall explain, we believe these results could help to see how spiral galaxies evolve into elliptical ones. A simple classical consequence of our isosceles triangle result gives a Keplerian type $4^{\textrm{th}}$ Law for 3-body problems. This is confined to the Appendix.

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A. Truman and R. Durran
Fri, 23 Sep 22
13/70

Comments: 43 pages, no figures

On the co-orbital asteroids in the solar system: medium-term timescale analysis of the quasi-coplanar objects [EPA]

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http://arxiv.org/abs/2209.05219


The focus of this work is the current distribution of asteroids in co-orbital motion with Venus, Earth and Jupiter, under a quasi-coplanar configuration and for a medium-term timescale of the order of 900 years. A co-orbital trajectory is a heliocentric orbit trapped in a 1:1 mean-motion resonance with a given planet. As such, to model it this work considers the Restricted Three-Body Problem in the circular-planar case with the help of averaging techniques. The domain of each co-orbital regime, that is, the quasi-satellite motion, the horseshoe motion and the tadpole motion, can be neatly defined by means of an integrable model and a simple bi-dimensional map, that is invariant with respect to the mass parameter of the planet, and turns out to be a remarkable tool to investigate the distribution of the co-orbitals objects of interest. The study is based on the data corresponding to the ephemerides computed by the JPL Horizons system for asteroids with a sufficient low orbital inclination with respect to the Sun-planet orbital plane. These objects are cataloged according to their current dynamics, together with the transitions that occur in the given time frame from a given type of co-orbital motion to another. The results provide a general catalog of co-orbital asteroids in the solar system, the first one to our knowledge, and an efficient mean to study transitions.

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S. Ruzza, A. Pousse and E. Alessi
Tue, 13 Sep 22
40/85

Comments: N/A

First post-Newtonian $N$-body problem in Einstein-Cartan theory with the Weyssenhoff fluid: equations of motion [CL]

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http://arxiv.org/abs/2208.09839


We derive the equations of motion for an $N$-body system in the Einstein-Cartan gravity theory at the first post-Newtonian order by exploiting the Weyssenhoff fluid as the spin model. Our approach consists in performing the point-particle limit of the continuous description of the gravitational source. The final equations provide a hint for the validity of the effacing principle at 1PN level in Einstein-Cartan model. The analogies with the general relativistic dynamics involving the macroscopic angular momentum are also discussed.

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E. Battista and V. Falco
Tue, 23 Aug 22
27/79

Comments: Paper features: 13 pages; 1 figure; 67 references. Accepted on EPJ C on the 20th of August 2022

A Comprehensive Perturbative Formalism for Phase-Mixing in Perturbed Disks. I. Phase spirals in an Infinite, Isothermal Slab [GA]

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http://arxiv.org/abs/2208.05038


Galactic disks are highly responsive systems that often undergo external perturbations and subsequent collisionless equilibration, predominantly via phase-mixing. We use linear perturbation theory to study the response of infinite isothermal slab analogues of disks to perturbations with diverse spatio-temporal characteristics. Without self-gravity of the response, the dominant Fourier modes that get excited in a disk are the bending and breathing modes, which, due to vertical phase-mixing, trigger local phase-space spirals that are one- and two-armed, respectively. We demonstrate how the lateral streaming motion of slab stars causes phase spirals to damp out over time. The ratio of the perturbation timescale ($\tau_{\mathrm{P}}$) to the local, vertical oscillation time ($\tau_z$) ultimately decides which of the two modes is excited. Faster, more impulsive ($\tau_{\mathrm{P}} < \tau_z$) and slower, more adiabatic ($\tau_{\mathrm{P}} > \tau_z$) perturbations excite stronger breathing and bending modes, respectively, although the response to very slow perturbations is exponentially suppressed. For encounters with satellite galaxies, this translates to more distant and more perpendicular encounters triggering stronger bending modes. We compute the direct response of the Milky Way disk to several of its satellite galaxies, and find that recent encounters with all of them excite bending modes in the Solar neighborhood. The encounter with Sagittarius triggers a response that is at least $1-2$ orders of magnitude larger than that due to any other satellite, including the Large Magellanic Cloud. We briefly discuss how ignoring the presence of a dark matter halo and the self-gravity of the response might impact our conclusions.

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U. Banik, M. Weinberg and F. Bosch
Thu, 11 Aug 22
49/68

Comments: Accepted for publication in ApJ; 7 figures, 1 table

The Disordered Heterogeneous Universe: Galaxy Distribution and Clustering Across Length Scales [CEA]

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http://arxiv.org/abs/2207.00519


Studies of disordered heterogeneous media and galaxy cosmology share a common goal: analyzing the distribution of particles at microscales' to predict physical properties atmacroscales’, whether for a liquid, composite material, or entire Universe. The former theory provides an array of techniques to characterize a wide class of microstructures; in this work, we apply them to the distributions of galaxies. We focus on the lower-order correlation functions, void' andparticle’ nearest-neighbor functions, pair-connectedness functions, percolation properties, and a scalar order metric. Compared to homogeneous Poisson and typical disordered systems, the cosmological simulations exhibit enhanced large-scale clustering and longer tails in the nearest-neighbor functions, due to the presence of quasi-long-range correlations. On large scales, the system appears hyperuniform', due to primordial density fluctuations, whilst on the smallest scales, the system becomes almostantihyperuniform’, and, via the order metric, is shown to be a highly correlated disordered system. Via a finite scaling analysis, we show that the percolation threshold of the galaxy catalogs is significantly lower than for Poisson realizations; this is consistent with the observation that the galaxy distribution contains larger voids. However, the two sets of simulations share a fractal dimension, implying that they lie in the same universality class. Finally, we consider the ability of large-scale clustering statistics to constrain cosmological parameters using simulation-based inference. Both the nearest-neighbor distribution and pair-connectedness function considerably tighten bounds on the amplitude of cosmological fluctuations at a level equivalent to observing twenty-five times more galaxies. These are a useful alternative to the three-particle correlation, and are computable in much reduced time. (Abridged)

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O. Philcox and S. Torquato
Mon, 4 Jul 22
30/62

Comments: 27 pages, submitted to Phys. Rev. X

After the Dark Ages [CEA]

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http://arxiv.org/abs/2205.11929


After recalling some puzzles in cosmology and briefly reviewing the Friedmann-Lema\^itre cosmos a simple unified model of the Dark Sector'' is described. This model involves a scalar field and a pseudo-scalar axion field that give rise to Dark Energy in the form ofquintessence” and to “fuzzy” Dark Matter, respectively. Predictions of the model concerning the late-time evolution of the Universe and possible implications for the problem of the observed Matter-Antimatter Asymmetry in the Universe are sketched.

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J. Fröhlich
Wed, 25 May 22
51/56

Comments: 17 pages (including bibliography), no figures

Instabilities Appearing in Effective Field theories: When and How? [CL]

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http://arxiv.org/abs/2205.01055


Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories (EFT) originated from cosmological studies. In particular, we are interested in the equation $\partial_t^2 u(x,t) = \alpha (\partial_x u(x,t))^2 +\beta \partial_x^2 u(x,t)$ in $1+1$ dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when $\alpha >0$. We study the detailed nature of this divergence as a function of the parameters $\alpha>0 $ and $\beta\ge0$. The divergence does not disappear even when $\beta $ is very large contrary to what one might believe. But it will take longer to appear as $\beta$ increases when $\alpha$ is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to $3+1$ dimensions.

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J. Eckmann, F. Hassani and H. Zaag
Tue, 3 May 22
51/82

Comments: 19 pages, 5 figures

A Birman-Schwinger Principle in General Relativity: Linearly Stable Shells of Collisionless Matter Surrounding a Black Hole [CL]

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http://arxiv.org/abs/2204.10620


We develop a Birman-Schwinger principle for the spherically symmetric, asymptotically flat Einstein-Vlasov system. It characterizes stability properties of steady states such as the positive definiteness of an Antonov-type operator or the existence of exponentially growing modes in terms of a one-dimensional variational problem for a Hilbert-Schmidt operator. This requires a refined analysis of the operators arising from linearizing the system, which uses action-angle type variables. For the latter, a single-well structure of the effective potential for the particle flow of the steady state is required. This natural property can be verified for a broad class of singularity-free steady states. As a particular example for the application of our Birman-Schwinger principle we consider steady states where a Schwarzschild black hole is surrounded by a shell of Vlasov matter. We prove the existence of such steady states and derive linear stability if the mass of the Vlasov shell is small compared to the mass of the black hole.

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S. Günther, G. Rein and C. Straub
Mon, 25 Apr 22
25/36

Comments: 58 pages, 1 figure

Gravitational Waves in ECSK theory: Robustness of mergers as standard sirens and nonvanishing torsion [CL]

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http://arxiv.org/abs/2204.00090


The amplitude propagation of gravitational waves in an Einstein-Cartan-Sciamma-Kibble (ECSK) theory is studied by assuming a dark matter spin tensor sourcing for spacetime torsion at cosmological scales. The analysis focuses on a weak-torsion regime, such that gravitational wave emission, at leading and subleading orders, does not deviate from standard General Relativity. We show that, in principle, the background torsion induced by an eventual dark matter spin component could lead to an anomalous dampening or amplification of the gravitational wave amplitude, after going across a long cosmological distance. The importance of this torsion-induced anomalous propagation of amplitude for binary black hole mergers is assessed. For realistic late-universe astrophysical scenarios, the effect is tiny and falls below detection thresholds, even for near-future interferometers such as LISA. To detect this effect may not be impossible, but it is still beyond our technological capabilities.

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E. Elizalde, F. Izaurieta, C. Riveros, et. al.
Mon, 4 Apr 22
26/50

Comments: 29 pages, 2 figures, comments welcome

How the Spirals in the Milky Way's ISM form [GA]

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http://arxiv.org/abs/2203.08672


We construct a model for the Milky Way where the interstellar medium (ISM) is equipped with self-consistent dynamics. In simulations a spiral structure emerges from this model that is almost identical with the one in the Milky Way’s ISM. Further, the Jeans instability offers an explanation for the observed velocity dispersion of atomic hydrogen in the ISM; this instability vanishes from our model if we choose a velocity dispersion just above the observed one. Surprisingly, our model gets along completely without dark matter. The ‘missing mass’ distributes uniformly over the baryonic components making it possible to explain the occurring mass gap.

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J. Frenkler
Thu, 17 Mar 22
10/66

Comments: 25 pages, 13 Figures, additional video material can be found on this https URL

On the best lattice quantizers [CL]

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http://arxiv.org/abs/2202.09605


A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization noise: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any locally optimal lattice quantizer and (ii) for an optimal product lattice, if the component lattices are themselves locally optimal. We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the Zador upper bound.

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E. Agrell and B. Allen
Thu, 24 Feb 22
47/52

Comments: N/A

A numerical criterion evaluating the robustness of planetary architectures; applications to the $\upsilon$ Andromedæ system [EPA]

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http://arxiv.org/abs/2202.08616


We revisit the problem of the existence of KAM tori in extrasolar planetary systems. Specifically, we consider the $\upsilon$ Andromed{\ae} system, by modelling it with a three-body problem. This preliminary study allows us to introduce a natural way to evaluate the robustness of the planetary orbits, which can be very easily implemented in numerical explorations. We apply our criterion to the problem of the choice of a suitable orbital configuration which exhibits strong stability properties and is compatible with the observational data that are available for the $\upsilon$ Andromed{\ae} system itself.

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U. Locatelli, C. Caracciolo, M. Sansottera, et. al.
Fri, 18 Feb 22
14/63

Comments: N/A

Analysis of performance and robustness against jitter of various search methods for acquiring optical links in space [CL]

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http://arxiv.org/abs/2202.00784


We discuss various methods for acquiring optical links in space using a dedicated acquisition sensor. Statistical models are developed and simple analytical equations derived that compare the performance between a single and dual spiral scan approach as well as between sequential and parallel acquisition of link chains. Simple derived analytical equations allow relating essential search parameters such as track width, variance of the uncertainty distribution, capture radius and scan speed to the probabilities of acquiring the links within a specific time. We also assess the probability of failing to acquire a link due to beam jitter and derive a simple analytical model that allows determining the maximum tolerable jitter for a given beam overlap and required probability of success. All results are validated by Monte Carlo simulations and applied to the concrete example of the GRACE FO mission.

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G. Hechenblaikner
Thu, 3 Feb 22
16/56

Comments: 11 pages, two column format, 9pt font

Exact general solutions for cosmological scalar field evolution in a background-dominated expansion [CL]

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http://arxiv.org/abs/2202.01132


We derive exact general solutions (as opposed to attractor particular solutions) and corresponding first integrals for the evolution of a scalar field $\phi$ in a universe dominated by a background fluid with equation of state parameter $w_B$. In addition to the previously-examined linear [$V(\phi) = V_0 \phi$] and quadratic [$V(\phi) = V_0 \phi^2$] potentials, we show that exact solutions exist for the power law potential $V(\phi) = V_0 \phi^n$ with $n = 4(1+w_B)/(1-w_B) + 2$ and $n = 2(1+w_B)/(1-w_B)$. These correspond to the potentials $V(\phi) = V_0 \phi^6$ and $V(\phi) = V_0 \phi^2$ for matter domination and $V(\phi) = V_0 \phi^{10}$ and $V(\phi) = V_0 \phi^4$ for radiation domination. The $\phi^6$ and $\phi^{10}$ potentials can yield either oscillatory or non-oscillatory evolution, and we use the first integrals to determine how the initial conditions map onto each form of evolution. The exponential potential yields an exact solution for a stiff/kination ($w_B = 1$) background. We use this exact solution to derive an analytic expression for the evolution of the equation of state parameter, $w_\phi$, for this case.

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R. Scherrer
Thu, 3 Feb 22
55/56

Comments: 13 pages, 3 figures, requests for citations to solutions previously appearing in the mathematics literature are welcome

Cooking pasta with Lie groups [CL]

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http://arxiv.org/abs/2201.12598


We extend the (gauged) Skyrme model to the case in which the global isospin group (which usually is taken to be $SU(N)$) is a generic compact connected Lie group $G$. We analyze the corresponding field equations in (3+1) dimensions from a group theory point of view. Several solutions can be constructed analytically and are determined by the embeddings of three dimensional simple Lie groups into $G$, in a generic irreducible representation. These solutions represent the so-called nuclear pasta state configurations of nuclear matter at low energy. We employ the Dynkin explicit classification of all three dimensional Lie subgroups of exceptional Lie group to classify all such solutions in the case $G$ is an exceptional simple Lie group, and give all ingredients to construct them explicitly. As an example, we construct the explicit solutions for $G=G_{2}$. We then extend our ansatz to include the minimal coupling of the Skyrme field to a $U(1)$ gauge field. We extend the definition of the topological charge to this case and then concentrate our attention to the electromagnetic case. After imposing a “free force condition” on the gauge field, the complete set of coupled field equations corresponding to the gauged Skyrme model minimally coupled to an Abelian gauge field is reduced to just one linear ODE keeping alive the topological charge. We discuss the cases in which such ODE belongs to the (Whittaker-)Hill and Mathieu types.

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S. Cacciatori, F. Canfora, M. Lagos, et. al.
Tue, 1 Feb 22
45/73

Comments: 51 pages, accepted for publication in NPB

Gauge invariant approach to nonmetricity theories and the second clock effect [CL]

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http://arxiv.org/abs/2201.03076


In this paper we discuss on recent attempts aimed at demonstrating that the second clock effect (SCE) does not take place in Weyl spaces, which is contrary to well-known results. These attempts include Weyl gauge theories of gravity, as well as the symmetric teleparallel theories (STTs). Our approach to this issue is based on the power of Weyl gauge symmetry (WGS) which is a manifest symmetry of the basic laws of Weyl geometry. Through proper consideration of WGS we shall show that the SCE, being an effect of purely geometric nature, does not depend on the chosen theory of gravity and matter. Quite the contrary, the SCE singles out those matter couplings which are phenomenologically compatible with the underlying geometric laws. Here we consider both, spacetimes based in Weyl geometry with arbitrary nonmetricity (generalized Weyl geometry), as well as, standard Weyl spaces where the nonmetricity is proportional to the product of a Weyl gauge vector by the metric. This issue is of special relevance for the fate of the STTs which are being intensively applied in the cosmological framework. As we shall show, if realize that WGS is a manifest symmetry of generalized Weyl spaces, neither the Weyl gauge theories nor the STTs are free of the second clock effect, unless Weyl integrable geometry (WIG) spaces are considered.

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I. Quiros
Tue, 11 Jan 22
29/95

Comments: 16 pages, no figures

Celestial Mechanics Solutions where the Future is a Perfect Reflection of the Past [CL]

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http://arxiv.org/abs/2112.11922


Newton’s equations of celestial mechanics are shown to possess a continuum of solutions in which the future trajectories of the N bodies are a perfect reflection of their past. These solutions evolve from zero initial velocities of the N bodies. Consequently, the future gravitational forces acting on the N bodies are also a perfect reflection of their past. The proof is carried out via Taylor series expansions. A perturbed system of equations of the N body problem is also considered. All real valued solutions of this perturbed system have no singularities on the real line. The perturbed system is shown to have a continuum of solutions that possess symmetry where the future velocities of the N bodies are a perfect reflection of their past. The positions and accelerations of the N bodies are then odd functions of the time. All N bodies then evolve from one location in space.

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A. Abdulhussein and H. Gingold
Thu, 23 Dec 21
13/63

Comments: N/A

Topological Quantification of the "Anemone" (Branching) Solar Flares [SSA]

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http://arxiv.org/abs/2111.06730


The so-called “anemone” solar flares are an interesting type of the space plasma phenomena, where multiple null points of the magnetic field are connected with each other and with the magnetic sources by the separators, thereby producing the complex branching configurations. Here, using the methods of dynamical systems and Morse-Smale theory, we derive a few universal topological relations between the numbers of the null points and sources of various kinds with arbitrary arrangement in the above-mentioned structures. Such relations can be a valuable tool both for a quantification of the already-observed anemone flares and for a prediction of the new ones in complex magnetic configurations.

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E. Zhuzhoma, V. Medvedev, Y. Dumin, et. al.
Mon, 15 Nov 21
32/52

Comments: LaTeX2e, elsarticle documentclass, 19 pages, 5 EPS figures

A new first-order formulation of the Einstein equations exploiting analogies with electrodynamics [CL]

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http://arxiv.org/abs/2111.05282


The Einstein and Maxwell equations are both systems of hyperbolic equations which need to satisfy a set of elliptic constraints throughout evolution. However, while electrodynamics (EM) and magnetohydrodynamics (MHD) have benefited from a large number of evolution schemes that are able to enforce these constraints and are easily applicable to curvilinear coordinates, unstructured meshes, or N-body simulations, many of these techniques cannot be straightforwardly applied to existing formulations of the Einstein equations. We develop a 3+1 a formulation of the Einstein equations which shows a striking resemblance to the equations of relativistic MHD and to EM in material media. The fundamental variables of this formulation are the frame fields, their exterior derivatives, and the Nester-Witten and Sparling forms. These mirror the roles of the electromagnetic 4-potential, the electromagnetic field strengths, the field excitations and the electric current. The role of the lapse function and shift vector, corresponds exactly to that of the scalar electric potential. The formulation, that we name dGREM (for differential forms, General Relativity and Electro-Magnetism), is manifestly first order and flux-conservative, which makes it suitable for high-resolution shock capturing schemes and finite-element methods. Being derived as a system of equations in exterior derivatives, it is directly applicable to any coordinate system and to unstructured meshes, and leads to a natural discretization potentially suitable for the use of machine-precision constraint propagation techniques such as the Yee algorithm and constrained transport. Due to these properties, we expect this new formulation to be beneficial in simulations of many astrophysical systems, such as binary compact objects and core-collapse supernovae as well as cosmological simulations of the early universe.

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H. Olivares, I. Peshkov, E. Most, et. al.
Wed, 10 Nov 21
8/63

Comments: 20 pages, 1 figure

Orbit determination with the Keplerian integrals [EPA]

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http://arxiv.org/abs/2111.02406


We review two initial orbit determination methods for too short arcs (TSAs) of optical observations of a solar system body. These methods employ the conservation laws of Kepler’s problem, and allow to attempt the linkage of TSAs referring to quite far epochs, differing by even more than one orbital period of the observed object. The first method ({\tt Link2}) concerns the linkage of 2 TSAs, and leads to a univariate polynomial equation of degree 9. An optimal property of this polynomial is proved using Gr\”obner bases theory. The second method ({\tt Link3}) is thought for the linkage of 3 TSAs, and leads to a univariate polynomial equation of degree 8. A numerical test is shown for both algorithms.

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G. Gronchi
Fri, 5 Nov 21
57/72

Comments: N/A

Scattering and absorption of a bosonic field impinging on a charged black hole in the Einstein-Maxwell-dilaton theory [CL]

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http://arxiv.org/abs/2111.01595


In this study, we revisit the absorption and scattering process by which a massless bosonic field impinges on a charged dilatonic black hole. Using the partial wave method, we determine numerically the total absorption cross-section in terms of the decoupling parameter called $M\omega$, finding that the amplitude of the dilatonic black hole is lower than the Reissner-Nordstr\”om one for mild frequencies. In the limit of high-frequency, the absorption cross-section exhibits two different kinds of complex behaviors, one is referred to as the fine structure and the other as the hyperfine structure. To fully grasp the main properties of the charged dilatonic black hole, we consider a different framework where the compact object is impinged by a charged massive bosonic field. In the low-frequency limit, we argue that the absorption cross-section presents two different phases, which are indicated by the value of a critical velocity. Depending on the dark matter model and the black hole mass, we show that both phases are relevant. We verify that the superradiance scattering takes place, being enhanced by smaller scalar field mass but larger values of the scalar field charge. For intermediate frequency, the superradiant effect is lessened in relation to the Reissner-Nordstr\”om case. However, this effect does not necessarily imply the existence of any instability. In order to trigger the superradiant instability, two conditions must be met. There must be unstable modes that remain trapped outside the event horizon with a mechanism based on the reflecting-mirror boundary conditions. This mechanism allows for the system composed of a charged scalar field plus a charged black hole to produce a charged black hole bomb. We provide an analytic formula (lower bound) for the values of the charge field which can trigger this superradiant instability.

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M. Richarte, &. Martins and J. Fabris
Wed, 3 Nov 21
71/106

Comments: 32 pages in two-columns format, 39 figures. Comments are welcome

Closed form perturbation theory in the restricted three-body problem without relegation [EPA]

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http://arxiv.org/abs/2110.14489


We propose a closed-form normalization method suitable for the study of the secular dynamics of small bodies in heliocentric orbits perturbed by the tidal potential of a planet with orbit external to the orbit of the small body. The method makes no use of relegation, thus, circumventing all convergence issues related to that technique. The method is based on a convenient use of a book-keeping parameter keeping simultaneously track of all the small quantities in the problem. The book-keeping affects both the Lie series and the Poisson structure employed in successive perturbative steps. In particular, it affects the definition of the normal form remainder at every normalization step. We show the results obtained by assuming Jupiter as perturbing planet and we discuss the validity and limits of the method.

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I. Cavallari and C. Efthymiopoulos
Thu, 28 Oct 21
72/76

Comments: N/A

Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part III: — Realization of exact solutions [CL]

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http://arxiv.org/abs/2110.13519


This is the Part III paper of our series of papers on a gauge-invariant perturbation theory on the Schwarzschild background spacetime. After reviewing our general framework of the gauge-invariant perturbation theory and the proposal on the gauge-invariant treatments for $l=0,1$ mode perturbations on the Schwarzschild background spacetime in [K.~Nakamura, arXiv:21XX.XXXXX], we examine the problem whether the $l=0,1$ even-mode solutions derived in the Part II paper [K.~Nakamura, arXiv:21XX.XXXXX] are physically reasonable, or not. We consider the linearized versions of the Lema\^itre-Tolman-Bondi solution and the C-metric. As the result, we show that our derived even-mode solutions to the linearized Einstein equations actually realize above two linearized solutions. This fact supports that our derived solutions are physically reasonable, which implies that our proposal on the gauge-invariant treatments for $l=0,1$ mode perturbations are also physically reasonable. We also briefly summarize our conclusions of our series of papers.

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K. Nakamura
Wed, 27 Oct 21
78/80

Comments: 38 pages, no figure, The Part III paper of the full paper version of the previous short papers arXiv:2102.083v3[gr-qc]; arXiv:2102.10650v3[gr-qc] (v1)

Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part I : — Formulation and odd-mode perturbations [CL]

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http://arxiv.org/abs/2110.13508


This is the Part I paper of our series of papers on a gauge-invariant perturbation theory on the Schwarzschild background spacetime. After reviewing our general framework of the gauge-invariant perturbation theory, we propose the strategy of the gauge-invariant perturbation theory on the Schwarzschild spacetime. In the above general framework, the “zero-mode problem” was a remaining important problem to develop gauge-invariant perturbation theories on generic background spacetime. In perturbation theories on the Schwarzschild background spacetime, $l=0,1$-mode problem corresponds to the above “zero-mode problem.” The above strategy proposed in this paper is a resolution of this $l=0,1$-mode problem in perturbations on the Schwarzschild background spacetime. Following this proposal, we derive the linearized Einstein equations for any modes of $l\geq 0$ in gauge-invariant manner. We discuss the solutions to the odd-mode perturbation equations in the linearized Einstein equations and show that these perturbations include the Kerr parameter perturbation in these odd-mode perturbation, which is physically reasonable. In the Part II and Part III papers [K.~Nakamura, arXiv:21XX.XXXXX; arXiv:21XX.XXXXX.] of this series of papers, we will show that the even-mode solutions to the linearized Einstein equations obtained through our proposal are also physically reasonable. Then, we conclude that our proposal of the resolution of the $l=0,1$-mode problem is also physically reasonable.

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K. Nakamura
Wed, 27 Oct 21
80/80

Comments: 52 pages, 3 figures, The Part I paper of the full paper version of the previous short papers arXiv:2102.083v3[gr-qc]; arXiv:2102.10650v3[gr-qc] (v1)

A Small Variation of the Circular Hodograph Theorem and the Best Elliptical Trajectory of the Planets [EPA]

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http://arxiv.org/abs/2109.11664


A small variation of the circular shape of the hodograph theorem states that for every elliptical solution of the two-body problem, it is possible to find an appropriate inertial frame such that the speed of the bodies is constant. We use this result and data from the NASA JPL Horizon Web Interface to find the best fitting ellipse for the trajectory of Mercury, Venus, Earth, Mars, and Jupiter. The process requires us to find procedures to obtain the plane and ellipse that best fit a collection of points in space. We show that if we aim for the plane that minimizes the sum of the square distances from the given points to the unknown plane, we obtain three planes that appear to divide the set of points equally into octants, one of these being our desired plane of best fit. We provide a detailed proof of the hodograph theorem.

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C. Cater, O. Perdomo and A. Valentine
Mon, 27 Sep 21
47/68

Comments: 22 pages, 9 figures

Einstein flow with matter sources: stability and convergence [CL]

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http://arxiv.org/abs/2108.12103


Two recent articles \cite{ashtekar2015general, moncrief2019could} suggested an interesting dynamical mechanism within the framework of the vacuum Einstein flow (or Einstein-$\Lambda$ flow if a positive cosmological constant $\Lambda$ is included) which suggests that many closed (compact without boundary) manifolds that do not support homogeneous and isotropic metrics \textit{at all} will nevertheless evolve to be asymptotically compatible with the observed approximate homogeneity and isotropy of the physical universe. These studies however did not include matter sources. Therefore the aim of the present study is to include suitable matter sources and investigate whether one is able to draw a similar conclusion.

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V. Moncrief and P. Mondal
Tue, 31 Aug 21
40/73

Comments: 20 pages. arXiv admin note: text overlap with arXiv:1903.00323, arXiv:1911.01233

Invariants in Polarimetric Interferometry: a non-Abelian Gauge Theory [CL]

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http://arxiv.org/abs/2108.11400


The discovery of magnetic fields close to the M87 black hole using Very Long Baseline Interferometry by the Event Horizon Telescope collaboration utilized the novel concept of “closure traces”, that are immune to antenna-based corruptions. We take a fundamentally new approach to this promising tool of polarimetric interferometry. The corruption of measurements of polarized signals at the individual antennas are represented by general $2\times 2$ complex matrices, which are identified with gauge transformations belonging to the group $\textrm{GL}(2,\mathbb{C})$, so the closure traces now appear as gauge-invariant quantities. We apply this formalism to polarimetric interferometry and generalize it to any number of interferometer elements. Our approach goes beyond existing studies in the following respects: (1) we do not need auto-correlations, which are susceptible to large systematic biases, and therefore unreliable (2) we use triangular combinations of correlations as basic building blocks (analogous to the “elementary plaquettes” of lattice gauge theory), and (3) we use the Lorentz group and its properties to transparently identify a complete and independent set of invariants. This set contains all the information immune to corruption available in the interferometer measurements, thus providing robust constraints which would be important in future interferometric studies.

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J. Samuel, R. Nityananda and N. Thyagarajan
Fri, 27 Aug 21
35/67

Comments: 9 pages (including references), 0 figures, submitted to Physical Review Letters. Contains appendices not included in the journal version

Invariants in Co-polar Interferometry: an Abelian Gauge Theory [IMA]

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http://arxiv.org/abs/2108.11399


An $N$-element interferometer measures correlations between pairs of array elements. Closure invariants associated with closed loops among array elements are immune to multiplicative, local, element-based corruptions that occur in these measurements. Till now, it has been unclear how a complete set of independent invariants can be analytically determined. We view the local, element-based corruptions in co-polar correlations as gauge tranformations belonging to the gauge group $\textrm{GL}(1,\mathbb{C})$. Closure quantities are then naturally gauge invariant. Using an Abelian $\textrm{GL}(1,\mathbb{C})$ gauge theory, we provide a simple and effective formalism to isolate the complete set of independent closure invariants from co-polar interferometric correlations only using quantities defined on the $(N-1)(N-2)/2$ elementary and independent triangular loops. The $(N-1)(N-2)/2$ closure phases and $N(N-3)/2$ closure amplitudes (totaling $N^2-3N+1$ real invariants), familiar in astronomical interferometry, naturally emerge from this formalism, which unifies what has required separate treatments until now. Our formalism does not require auto-correlations, but can easily include them if reliably measured, including potentially from cross-correlation between two short-spaced elements. The gauge theory framework presented here extends to $\textrm{GL}(2,\mathbb{C}$) for full polarimetric interferometry presented in a companion paper, which generalizes and clarifies earlier work. Our findings can be relevant to cutting-edge co-polar and full polarimetric very long baseline interferometry measurements to determine features very near the event horizons of blackholes at the centers of M87, Centaurus~A, and the Milky Way.

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N. Thyagarajan, R. Nityananda and J. Samuel
Fri, 27 Aug 21
66/67

Comments: 10 pages (including references), 0 figures, submitted to Physical Review D

Integrable modified gravity cosmological models with an additional scalar field [CL]

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http://arxiv.org/abs/2108.10276


We consider modified gravity cosmological models that can be transformed into two-field chiral cosmological models by the conformal metric transformation. For the $R^2$ gravity model with an additional scalar field and the corresponding two-field model with the cosmological constant and nonstandard kinetic part of the action, the general solutions have been obtained in the spatially flat FLRW metric. We analyze the correspondence of the cosmic time solutions obtained and different possible evolutions of the Hubble parameters in the Einstein and Jordan frames.

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V. Ivanov and S. Vernov
Tue, 24 Aug 21
70/76

Comments: 21 pages, 3 figures

Master Functions and Equations for Vacuum Spherically-Symmetric Spacetimes [CL]

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http://arxiv.org/abs/2108.08668


Perturbation theory of vacuum spherically-symmetric spacetimes is a crucial tool to understand the dynamics of black hole perturbations. Spherical symmetry allows for an expansion of the perturbations in scalar, vector, and tensor harmonics. The resulting perturbative equations are decoupled for modes with different parity and different harmonic numbers. Moreover, for each harmonic and parity, the equations for the perturbations can be decoupled in terms of (gauge-invariant) master functions that satisfy 1+1 wave equations. By working in a completely general perturbative gauge, in this paper we study what is the most general master function that is linear in the metric perturbations and their first-order derivatives and satisfies a wave equation with a potential. The outcome of the study is that for each parity we have two branches of solutions with similar features. One of the branches includes the known results: In the odd-parity case, the most general master function is an arbitrary linear combination of the Regge-Wheeler and the Cunningham-Price-Moncrief master functions whereas in the even-parity case it is an arbitrary linear combination of the Zerilli master function and another master function that is new to our knowledge. The other branch is very different since it includes an infinite collection of potentials which in turn lead to an independent collection master of functions which depend on the potential. The allowed potentials satisfy a non-linear ordinary differential equation. Finally, all the allowed master functions are gauge invariant and can be written in a fully covariant form.

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M. Lenzi and C. Sopuerta
Fri, 20 Aug 21
38/59

Comments: 22 pages, revtex 4-2

MCMC generation of cosmological fields far beyond Gaussianity [CEA]

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http://arxiv.org/abs/2107.05639


Structure formation in our Universe creates non-Gaussian random fields that will soon be observed over almost the entire sky by the Euclid satellite, the Vera-Rubin observatory, and the Square Kilometre Array. An unsolved problem is how to analyze such non-Gaussian fields best, e.g. to infer the physical laws that created them. This problem could be solved if a parametric non-Gaussian sampling distribution for such fields were known, as this distribution could serve as likelihood during inference. We therefore create a sampling distribution for non-Gaussian random fields. Our approach is capable of handling strong non-Gaussianity, while perturbative approaches such as the Edgeworth expansion cannot. To imitate cosmological structure formation, we enforce our fields to be (i) statistically isotropic, (ii) statistically homogeneous, and (iii) statistically independent at large distances. We generate such fields via a Monte Carlo Markov Chain technique and find that even strong non-Gaussianity is not necessarily visible to the human eye. We also find that sampled marginals for pixel pairs have an almost generic Gauss-like appearance, even if the joint distribution of all pixels is markedly non-Gaussian. This apparent Gaussianity is a consequence of the high dimensionality of random fields. We conclude that vast amounts of non-Gaussian information can be hidden in random fields that appear nearly Gaussian in simple tests, and that it would be short-sighted not to try and extract it.

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J. Braspenning and E. Sellentin
Wed, 14 Jul 21
54/67

Comments: For submission and review to the Open Journal of Astrophysics

Revisiting the averaged problem in the case of mean-motion resonances of the restricted three-body problem. Global rigorous treatment and application to the co-orbital motion [EPA]

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http://arxiv.org/abs/2106.14810


A classical approach to the restricted three-body problem is to analyze the dynamics of the massless body in the synodic reference frame. A different approach is represented by the perturbative treatment: in particular the averaged problem of a mean-motion resonance allows to investigate the long-term behavior of the solutions through a suitable approximation that focuses on a particular region of the phase space. In this paper, we intend to bridge a gap between the two approaches in the specific case of mean-motion resonant dynamics, establish the limit of validity of the averaged problem, and take advantage of its results in order to compute trajectories in the synodic reference frame. After the description of each approach, we develop a rigorous treatment of the averaging process, estimate the size of the transformation and prove that the averaged problem is a suitable approximation of the restricted three-body problem as long as the solutions are located outside the Hill’s sphere of the secondary. In such a case, a rigorous theorem of stability over finite but large timescales can be proven. We establish that a solution of the averaged problem provides an accurate approximation of the trajectories on the synodic reference frame within a finite time that depend on the minimal distance to the Hill’s sphere of the secondary. The last part of this work is devoted to the co-orbital motion (i.e., the dynamics in 1:1 mean-motion resonance) in the circular-planar case. In this case, an interpretation of the solutions of the averaged problem in the synodic reference frame is detailed and a method that allows to compute co-orbital trajectories is displayed.

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A. Pousse and E. Alessi
Tue, 29 Jun 21
52/101

Comments: 24 pages, 7 figures, 1 table

Applications of Cosmological Perturbation Theory in the Late Universe [CEA]

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http://arxiv.org/abs/2106.10181


In this thesis, we discuss some of the applications of cosmological perturbation theory in the late universe. We begin by reviewing the tools used to understand the standard model of cosmology theoretically and to compute its observational consequences, including a detailed exposition of cosmological perturbation theory. We then describe the results in this thesis; we present novel analytical solutions for linear-order gravitational waves or tensor perturbations in a flat Friedmann-Robertson-Walker universe containing two perfect fluids — radiation and pressureless dust — and allowing for neutrino anisotropic stress. One of the results applies to any sub-horizon gravitational wave in such a universe. Another result applies to gravitational waves of primordial origin (for example, produced during inflation) and works both before and after they cross the horizon. These results improve on analytical approximations previously set out in the literature. Comparison with numerical solutions shows that both these approximations are accurate to within 1% or better, for a wide range of wave-numbers relevant for cosmology. We present a new and independent approach to computing the relativistic galaxy number counts to second order in cosmological perturbation theory. We also derive analytical expressions for the full second-order relativistic observed redshift, for the angular diameter distance and the volume spanned by a survey. We then compare our result with previous works which compute the general distance-redshift relation, finding that our result is in agreement at linear and leading nonlinear order. Lastly, we briefly study a class of almost scale-invariant Gauss-Bonnet modified gravity theory and derive the Einstein-like field equations to first order in cosmological perturbation theory in longitudinal gauge.

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J. Fuentes
Mon, 21 Jun 21
40/54

Comments: 149 pages, 11 figures, Doctoral Thesis. arXiv admin note: text overlap with arXiv:1911.08313 by other authors

Solutions of the imploding shock problem in a medium with varying density [CL]

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http://arxiv.org/abs/2106.04971


We consider the solutions of the Guderley problem, consisting of an imploding strong shock wave in an ideal gas with a power law initial density profile. The self-similar solutions, and specifically the similarity exponent which determines the behavior of the accelerating shock, are studied in detail, for cylindrical and spherical symmetries and for a wide range of the adiabatic index and the spatial density exponent. We then demonstrate how the analytic solutions can be reproduced in Lagrangian hydrodynamic codes, thus demonstrating their usefulness as a code validation and verification test problem.

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I. Giron, S. Balberg and M. Krief
Thu, 10 Jun 21
26/77

Comments: N/A

Collisionless equilibria in general relativity: stable configurations beyond the first binding energy maximum [CL]

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http://arxiv.org/abs/2105.05556


We numerically study the stability of collisionless equilibria in the context of general relativity. More precisely, we consider the spherically symmetric, asymptotically flat Einstein-Vlasov system in Schwarzschild and in maximal areal coordinates. Our results provide strong evidence against the well-known binding energy hypothesis which states that the first local maximum of the binding energy along a sequence of isotropic steady states signals the onset of instability. We do however confirm the conjecture that steady states are stable at least up to the first local maximum of the binding energy. For the first time, we observe multiple stability changes for certain models. The equations of state used are piecewise linear functions of the particle energy and provide a rich variety of different equilibria.

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S. Günther, C. Straub and G. Rein
Thu, 13 May 21
39/60

Comments: 23 pages, 8 figures

Timescales of the chaos onset in the general relativistic Poynting-Robertson effect [CL]

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http://arxiv.org/abs/2105.00965


It has been proved that the general relativistic Poynting-Robertson effect in the equatorial plane of Kerr metric shows a chaotic behavior for a suitable range of parameters. As a further step, we calculate the timescale for the onset of chaos through the Lyapunov exponents, estimating how this trend impacts on the observational dynamics. We conclude our analyses with a discussion on the possibility to observe this phenomenon in neutron star and black hole astrophysical sources.

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V. Falco and W. Borrelli
Tue, 4 May 21
14/72

Comments: 11 pages; 4 figures; 3 tables; accepted for publication on PRD

An Exact Integral-to-Sum Relation for Products of Bessel Functions [CL]

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http://arxiv.org/abs/2104.10169


A useful identity relating the infinite sum of two Bessel functions to their infinite integral was discovered in Dominici et al. (2012). Here, we extend this result to products of $N$ Bessel functions, and show it can be straightforwardly proven using the Abel-Plana theorem. For $N=2$, the proof is much simpler than that of the former work, and significantly enlarges the range of validity.

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O. Philcox and Z. Slepian
Thu, 22 Apr 2021
2/44

Comments: 13 pages, 1 figure. Comments welcome

The optimal lattice quantizer in nine dimensions [CL]

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http://arxiv.org/abs/2104.10107


The optimal lattice quantizer is the lattice which minimizes the (dimensionless) second moment $G$. In dimensions $1$ to $8$, it has been proven that the optimal lattice quantizer is one of the classical lattices, or there is good numerical evidence for this. In contrast, more than two decades ago, convincing numerical studies showed that in dimension $9$, a non-classical lattice is optimal. The structure and properties of this lattice depend upon a single positive real parameter $a$, whose value was only known approximately. Here, for $a^2 < 1/2$, we give an exact analytic description of this one-parameter family of lattices and their Voronoi cells, and calculate their second moment, which is a $19$th order polynomial in $a$. This allows us to determine the exact value of $a$ which minimizes $G$. It is an algebraic number, defined by the root of a $9$th order polynomial, with $a \approx 0.573223794$. We also show that for this value of $a$, the covariance matrix (second moment tensor) is proportional to the identity, consistent with a theorem of Zamir and Feder for optimal quantizers. The same method can be used for arbitrary one-parameter families of laminated lattices, so may provide a useful tool to identify optimal quantizers in other dimensions as well.

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B. Allen and E. Agrell
Wed, 21 Apr 2021
55/72

Comments: 9 pages, 3 figures, face catalog attached as supplementary materials

Spatio-temporal linear instability analysis for arbitrary dispersion relations on the Lefschetz thimble in multidimensional spacetime [CL]

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http://arxiv.org/abs/1912.11177


In linear stability analysis of field quantities described by partial differential equations, the well-established classical theory is all but impossible to apply to concrete problems in its entirety even for uniform backgrounds when the spatial dimension is more than 1. In this study, using the Lefschetz thimble method, we develop a new formalism to give an explicit expression to the asymptotic behavior of linear perturbations. It is not only more mathematically rigorous than the previous theory but also useful practically in its applications to realistic problems, and, as such, has an impact on broad subjects in physics.

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T. Morinaga and S. Yamada
Wed, 31 Mar 2021
62/62

Comments: 10 pages, 18 figures, accepted for publication in Physical Review Research

Surface Gravity of Rotating Dumbbell Shapes [EPA]

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http://arxiv.org/abs/2102.11990


We investigate the problem of determining the shape of a rotating celestial object – e.g., a comet or an asteroid – under its own gravitational field. More specifically, we consider an object symmetric with respect to one axis – such as a dumbbell – that rotates around a second axis perpendicular to the symmetry axis. We assume that the object can be modeled as an incompressible fluid of constant mass density, which is regarded as a first approximation of an aggregate of particles.
In the literature, the gravitational field of a body is often described as a multipolar expansion involving spherical coordinates (Kaula, 1966). In this work we describe the shape in terms of cylindrical coordinates, which are most naturally adapted to the symmetry of the body, and we express the gravitational potential generated by the rotating body as a simple formula in terms of elliptic integrals. An equilibrium shape occurs when the gravitational potential energy and the rotational kinetic energy at the surface of the body balance each other out. Such an equilibrium shape can be derived as a solution of an optimization problem, which can be found via the variational method. We give an example where we apply this method to a two-parameter family of dumbbell shapes, and find approximate numerical solutions to the corresponding optimization problem.

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W. Lam, M. Gidea and F. Zypman
Thu, 25 Feb 21
6/50

Comments: N/A

On the existence of linearly oscillating galaxies [CL]

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http://arxiv.org/abs/2102.11672


We consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically.

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M. Hadzic, G. Rein and C. Straub
Wed, 24 Feb 21
42/64

Comments: 104 pages

Optimal Template Banks [IMA]

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http://arxiv.org/abs/2102.11254


When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,…) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space. The computational cost is proportional to the number of these points, and if that is fixed, the question arises, where should the points be placed in parameter space? The current literature advocates selecting the set of points (called a “template bank”) whose Wigner-Seitz (also called Voronoi) cells have the smallest covering radius ($\equiv$ smallest maximal mismatch). Mathematically, such a template bank is said to have “minimum thickness”. Here, we show that at fixed computational cost, for realistic populations of signal sources, the minimum thickness template bank does NOT maximize the expected number of detections. Instead, the most detections are obtained for a bank which minimizes a particular functional of the mismatch. For closely spaced templates, the most detections are obtained for a template bank which minimizes the average squared distance from the nearest template, i.e., the average expected mismatch. Mathematically, such a template bank is said to be the “optimal quantizer”. We review the optimal quantizers for template banks that are built as $n$-dimensional lattices, and show that even the best of these offer only a marginal advantage over template banks based on the humble cubic lattice.

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B. Allen
Tue, 23 Feb 21
24/79

Comments: 1 Figure

A Brief Introduction to the Adomian Decomposition Method, with Applications in Astronomy and Astrophysics [IMA]

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http://arxiv.org/abs/2102.10511


The Adomian Decomposition Method (ADM) is a very effective approach for solving broad classes of nonlinear partial and ordinary differential equations, with important applications in different fields of applied mathematics, engineering, physics and biology. It is the goal of the present paper to provide a clear and pedagogical introduction to the Adomian Decomposition Method and to some of its applications. In particular, we focus our attention to a number of standard first-order ordinary differential equations (the linear, Bernoulli, Riccati, and Abel) with arbitrary coefficients, and present in detail the Adomian method for obtaining their solutions. In each case we compare the Adomian solution with the exact solution of some particular differential equations, and we show their complete equivalence. The second order and the fifth order ordinary differential equations are also considered. An important extension of the standard ADM, the Laplace-Adomian Decomposition Method is also introduced through the investigation of the solutions of a specific second order nonlinear differential equation. We also present the applications of the method to the Fisher-Kolmogorov second order partial nonlinear differential equation, which plays an important role in the description of many physical processes, as well as three important applications in astronomy and astrophysics, related to the determination of the solutions of the Kepler equation, of the Lane-Emden equation, and of the general relativistic equation describing the motion of massive particles in the spherically symmetric and static Schwarzschild geometry.

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M. Mak, C. Leung and T. Harko
Tue, 23 Feb 21
58/79

Comments: 41 pages, no figures, accepted for publication in the Romanian Astronomical Journal

Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid [CL]

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http://arxiv.org/abs/2101.01696


In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain $\mathbb{T}\times \mathbb{R}$. In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their $L^2$ norm grows as $t^{1/2}$ and this confirms previous observations in the physics literature. Instead, the solenoidal component of the velocity field experience inviscid damping, meaning that it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order $\nu^{-1/6}$ (with $\nu^{-1}$ being proportional to the Reynolds number) on a time-scale $\nu^{-1/3}$, after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible fluid, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.

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P. Antonelli, M. Dolce and P. Marcati
Wed, 6 Jan 21
56/82

Comments: 39 pages. A preliminary analysis of the inviscid problem already appeared in our unpublished note arxiv.org/abs/2003.01694

Cosmology Without Windows: Quadratic Estimators for the Galaxy Power Spectrum [CEA]

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http://arxiv.org/abs/2012.09389


Conventional algorithms for galaxy power spectrum estimation measure the true spectrum convolved with a survey window function, which, for parameter inference, must be compared with a similarly convolved theory model. In this work, we directly estimate the unwindowed power spectrum multipoles using quadratic estimators akin to those introduced in the late 1990s. Under Gaussian assumptions, these are optimal and free from the leading-order effects of pixellization and non-Poissonian shot-noise. They may be straightforwardly computed given the survey data-set and a suite of simulations of known cosmology. We implement the pixel-based maximum-likelihood estimator and a simplification based on the FKP weighting scheme, both of which can be computed via FFTs and conjugate gradient descent methods. Furthermore, the estimators allow direct computation of spectrum coefficients in an arbitrary linear compression scheme, without needing to first bin the statistico. Applying the technique to a subset of the BOSS DR12 galaxies, we find that the pixel-based quadratic estimators give statistically consistent power spectra, compressed coefficients, and cosmological parameters to those obtained with the usual windowed approaches. Due to the sample’s low number density and compact window function, the optimal weighting scheme gives little improvement over the simplified form; this may change for dense surveys or those focusing on primordial non-Gaussianity. The technique is shown to be efficient and robust, and shows significant potential for measuring the windowless power spectrum and bispectrum in the presence of weak non-Gaussianity.

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O. Philcox
Fri, 18 Dec 20
24/78

Comments: 20+8 pages, 12 figures, submitted to Phys. Rev. D

Two-body neutral Coulomb system in a magnetic field at rest: from Hydrogen atom to positronium [CL]

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http://arxiv.org/abs/2012.00044


A simple uniform approximation for the nodeless wavefunction is constructed for a {\it neutral} system of two Coulomb charges of different masses $(-q,m_1)$ and $(q,m_2)$ at rest in a constant uniform magnetic field for the states of positive and negative parity, ${(1s_0)}$ and ${(2p_0)}$, respectively. It is shown that by keeping the mass and charge of one of the bodies fixed, all systems with different second body masses are related. This allows one to consider the second body as infinitely-massive and to take such a system as basic. Three physical systems are considered in details: the Hydrogen atom with (in)-finitely massive proton (deuteron, triton) and the positronium atom $(-e,e)$. We derive the Riccati-Bloch and Generalized-Bloch equations, which describe the domains of small and large distances, respectively. Based on the interpolation of the small and large distance behavior of the logarithm of the wavefunction, a compact 10-parametric function is proposed. Taken as a variational trial function it provides accuracy of not less than 6 significant digits (s.d.) ($\lesssim 10^{-6}$ in relative deviation) for the total energy in the whole domain of considered magnetic fields $[0\,,\,10^4]$ a.u. and not less than 3 s.d. for the quadrupole moment $Q_{zz}$. In order to get reference points the Lagrange Mesh Method with 16K mesh points was used to get from 10 to 6 s.d. in energy from small to large magnetic fields. Based on the Riccati-Bloch equation the first 100 perturbative coefficients for the energy, in the form of rational numbers, are calculated and, using the Pad\’e-Borel re-summation procedure, the energy is found with not less than 10 s.d. at magnetic fields $\leq 1$\,a.u.

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J. Valle, A. Turbiner and A. Ruiz
Wed, 2 Dec 20
51/71

Comments: 47 pages, 7 tables, 5 figures, 3 appendices

Transits close to the Lagrangian solutions $L_1,L_2$ in the Elliptic Restricted Three-body Problem [EPA]

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http://arxiv.org/abs/2011.14957


In the last decades a peculiar family of solutions of the Circular Restricted Three Body Problem has been used to explain the temporary captures of small bodies and spacecrafts by a planet of the Solar System. These solutions, which transit close to the Lagrangian points $L_1,L_2$ of the CRTBP, have been classified using the values of approximate local integrals and of the Jacobi constant. The use for small bodies of the Solar System requires to consider a hierarchical extension of the model, from the CRTBP to the the full $N$ planetary problem. The Elliptic Restricted Three Body, which is the first natural extension of the CRTBP, represents already a challenge, since global first integrals such as the Jacobi constant are not known for this problem. In this paper we extend the classification of the transits occurring close to the Lagrangian points $L_1,L_2$ of the ERTBP using a combination of the Floquet theory and Birkhoff normalizations. Provided that certain non-resonance conditions are satisfied, we conjugate the Hamiltonian of the problem to an integrable normal form Hamiltonian with remainder, which is used to define approximate local first integrals and to classify the transits of orbits through a neighbourhood of the Lagrange equilibria according to the values of these integrals. We provide numerical demonstrations for the Earth-Moon ERTBP.

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R. Páez and M. Guzzo
Tue, 1 Dec 20
36/108

Comments: 36 pages, 8 figures. Submitted for publication

M-theory, Black Holes and Cosmology [CL]

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http://arxiv.org/abs/2009.11339


This paper is dedicated to Mike Duff on the occasion of his 70th birthday. I discuss some issues of M-theory/string theory/supergravity closely related to Mike’s interests. I describe a relation between STU black hole entropy, Cayley hyperdeterminant, Bhargava cube and a 3-qubit Alice, Bob, Charlie triality symmetry. I shortly describe my recent work with Gunaydin, Linde, Yamada on M-theory cosmology, inspired by the work of Duff with Ferrara and Borsten, Levay, Marrani et al. Here we have 7-qubits, a party including Alice, Bob, Charlie, Daisy, Emma, Fred, George. Octonions and Hamming error correcting codes are at the base of these models. They lead to 7 benchmark targets of future CMB missions looking for primordial gravitational wave from inflation. I also show puzzling relations between the fermion mass eigenvalues in these cosmological models, exceptional Jordan eigenvalue problem, and black hole entropy. The symmetry of our cosmological models is illustrated by beautiful pictures of a Coxeter projection of the root system of E7.

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R. Kallosh
Fri, 25 Sep 20
-1825/62

Comments: 19 pages, 8 figures

Dispersion Relations in $κ$-Noncommutative Cosmology [CL]

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http://arxiv.org/abs/2009.01051


We study noncommutative deformations of the wave equation in curved backgrounds and discuss the modification of the dispersion relations due to noncommutativity combined with curvature of spacetime. Our noncommutative differential geometry approach is based on Drinfeld twist deformation, and can be implemented for any twist and any curved background. We discuss in detail the Jordanian twist $-$giving $\kappa$-Minkowski spacetime in flat space$-$ in the presence of a Friedman-Lema\^{i}tre-Robertson-Walker (FLRW) cosmological background. We obtain a new expression for the variation of the speed of light, depending linearly on the ratio $E_{ph}/E_{LV}$ (photon energy / Lorentz violation scale), but also linearly on the cosmological time, the Hubble parameter and inversely proportional to the scale factor.

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P. Aschieri, A. Borowiec and A. Pachoł
Fri, 4 Sep 20
-1375/65

Comments: 20 pages

Lukash plane waves, revisited [CL]

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http://arxiv.org/abs/2008.07801


The Lukash metric is a homogeneous gravitational wave which approximates at late times the behaviour of a generic class of spatially homogenous cosmological models with monotonically decreasing energy density. Following pioneering work of Siklos, we provide a self-contained account of the geometry and global structure of the spacetime. The latter contains a Killing horizon to the future of which the spacetime resembles an anisotropic version of the Milne cosmology and to the past of which it resemble the Rindler wedge. We discuss the Unruh effect whereby, for a suitable vacuum state, a class of uniformly accelerated observers experience a background of thermal radiation.

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M. Elbistan, P. Zhang, G. Gibbons, et. al.
Tue, 25 Aug 20
-1146/99

Comments: 29 pges, 3 figures

Application of the Gauss-Bonnet theorem to lensing in the NUT metric [CL]

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http://arxiv.org/abs/2008.10093


We show with the help of Fermat’s principle that every lightlike geodesic in the NUT metric projects to a geodesic of a two-dimensional Riemannian metric which we call the optical metric. The optical metric is defined on a (coordinate) cone whose opening angle is determined by the impact parameter of the lightlike geodesic. We show that, surprisingly, the optical metrics on cones with different opening angles are locally (but not globally) isometric. With the help of the Gauss-Bonnet theorem we demonstrate that the deflection angle of a lightlike geodesic is determined by an area integral over the Gaussian curvature of the optical metric. A similar result is known to be true for static and spherically symmetric spacetimes. The generalisation to the NUT spacetime, which is neither static nor spherically symmetric (at least not in the usual sense), is rather non-trivial.

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M. Halla and V. Perlick
Tue, 25 Aug 20
-1134/99

Comments: N/A

Ejection-collision orbits in two degrees of freedom problems in celestial mechanics [CL]

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http://arxiv.org/abs/2008.06526


In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of $N$-body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the bodies are involved. Regularizing the singularities, the total collision transforms into a 2-dimensional invariant manifold. The goal of this paper is to prove the existence of different types of ejection-collision orbits, that is, orbits that start and end at total collision. Such orbits are regarded as heteroclinic connections between two equilibrium points and are mainly characterized by the partial collisions that the trajectories find on their way. The proof of their existence is based on the transversality of 2-dimensional invariant manifolds and on the behavior of the dynamics on the total collision manifold, both of them are thoroughly described.

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M. Alvarez-Ramírez, E. Barrabés, M. Medina, et. al.
Tue, 18 Aug 20
-998/70

Comments: N/A

Dynamical and static solutions to $R=0$-scalar-tensor theory [CL]

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http://arxiv.org/abs/2007.11023


We consider the most cosmologically interesting and relevant case of scalar-tensor theory (STT) and derive new normal and phantom, dynamical and static, solutions. We determine the Bianchi I Kasner exponents and show that the dynamical solutions are heteroclinic orbits connecting two singularities. Approaching the singularities, a purely transverse expansion (no radial expansion or collapse) may occur.

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M. Azreg-Aïnou
Thu, 23 Jul 20
-449/83

Comments: 7 twocolumn pages

The invariance of the diffusion coefficient with the iterative operations of charged particles' transport equation [CL]

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http://arxiv.org/abs/2006.01706


The Spatial Parallel Diffusion Coefficient (SPDC) is one of the important quantities describing energetic charged particle transport. There are three different definitions for the SPDC, i.e., the Displacement Variance definition $\kappa_{zz}^{DV}=\lim_{t\rightarrow t_{\infty}}d\sigma^2/(2dt)$, the Fick’s Law definition $\kappa_{zz}^{FL}=J/X$ with $X=\partial{F}/\partial{z}$, and the TGK formula definition $\kappa_{zz}^{TGK}=\int_0^{\infty}dt \langle v_z(t)v_z(0) \rangle$. For constant mean magnetic field, the three different definitions of the SPDC give the same result. However, for focusing field it is demonstrated that the results of the different definitions are not the same. In this paper, from the Fokker-Planck equation we find that different methods, e.g., the general Fourier expansion and perturbation theory, can give the different Equations of the Isotropic Distribution Function (EIDFs). But it is shown that one EIDF can be transformed into another by some Derivative Iterative Operations (DIOs). If one definition of the SPDC is invariant for the DIOs, it is clear that the definition is also an invariance for different EIDFs, therewith it is an invariant quantity for the different Derivation Methods of EIDF (DMEs). For the focusing field we suggest that the TGK definition $\kappa_{zz}^{TGK}$ is only the approximate formula, and the Fick’s Law definition $\kappa_{zz}^{FL}$ is not invariant to some DIOs. However, at least for the special condition, in this paper we show that the definition $\kappa_{zz}^{DV}$ is the invariant quantity to the kinds of the DIOs. Therefore, for spatially varying field the displacement variance definition $\kappa_{zz}^{DV}$, rather than the Fick’s law definition $\kappa_{zz}^{FL}$ and TGK formula definition $\kappa_{zz}^{TGK}$, is the most appropriate definition of the SPDCs.

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J. Wang and G. Qin
Wed, 3 Jun 20
78/83

Comments: N/A

Maximal extension of the Schwarzschild metric: From Painlevé-Gullstrand to Kruskal-Szekeres [CL]

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http://arxiv.org/abs/2005.14211


We find a specific coordinate system that goes from the Painlev\’e-Gullstrand partial extension to the Kruskal-Szekeres maximal extension and thus exhibit the maximal extension of the Schwarzschild metric in a unified picture. We do this by adopting two time coordinates, one being the proper time of a congruence of outgoing timelike geodesics, the other being the proper time of a congruence of ingoing timelike geodesics, both parameterized by the same energy per unit mass $E$. $E$ is in the range $1\leq E<\infty$ with the limit $E=\infty$ yielding the Kruskal-Szekeres maximal extension. So, through such an integrated description one sees that the Kruskal-Szekeres solution belongs to this family of extensions parameterized by $E$. Our family of extensions is different from the Novikov-Lema\^itre family parameterized also by the energy $E$ of timelike geodesics, with the Novikov extension holding for $0<E<1$ and being maximal, and the Lema\^itre extension holding for $1\leq E<\infty$ and being partial, not maximal, and moreover its $E=\infty$ limit evanescing in a Minkowski spacetime rather than ending in the Kruskal-Szekeres spacetime.

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J. Lemos and D. Silva
Mon, 1 Jun 20
38/50

Comments: 18 pages, 7 figures

Emergence of classical behavior in the early universe [CL]

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http://arxiv.org/abs/2004.10684


We investigate three issues that have been discussed in the context of inflation: Fading of the importance of quantum non-commutativity; the phenomenon of quantum squeezing; and the ability to approximate the quantum state by a distribution function on the classical phase space. In the standard treatments, these features arise from properties of mode functions of quantum fields in (near) de Sitter space-time. Therefore, the three notions are often assumed to be essentially equivalent, representing different facets of the same phenomenon. We analyze them in general Friedmann-Lemaitre- Robertson-Walker space-times, through the lens of geometric structures on the classical phase space. The analysis shows that: (i) inflation does not play an essential role; classical behavior can emerge much more generally; (ii) the three notions are conceptually distinct; classicality can emerge in one sense but not in another; and, (iii) the third notion is realized in a surprisingly strong sense; there is exact equality between completely general $n$-point functions in the classical theory and those in the quantum theory, provided the quantum operators are Weyl ordered. These features arise already for linear cosmological perturbations by themselves: considerations such as mode-mode coupling, decoherence, and measurement theory –although important in their own right– are not needed for emergence of classical behavior in any of the three senses discussed. Generality of the results stems from the fact that they can be traced back to geometrical structures on the classical phase space, available in a wide class of systems. Therefore, this approach may also be useful in other contexts.

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A. Ashtekar, A. Corichi and A. Kesavan
Thu, 23 Apr 20
11/45

Comments: 38 pages; 3 Figures. The first and the last sections provide a succinct summary of the motivation results

Analytical solution of the Colombo top problem [EPA]

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http://arxiv.org/abs/2003.14198


The Colombo top is a basic model in the rotation dynamics of a celestial body moving on a precessing orbit and perturbed by a gravitational torque. The paper presents a detailed study of analytical solution to this problem. By solving algebraic equations of degree 4, we provide the expressions for the extreme points of trajectories as functions of their energy. The location of stationary points (known as the Cassini states) is found as the function of the two parameters of the problem. Analytical solution in terms the Weierstrass and the Jacobi elliptic functions is given for regular trajectories. Some trajectories are expressible through elementary functions: not only the homoclinic orbits, as expected, but also a special periodic solution whose energy is equal to that of the first Cassini state (unnoticed in previous studies).

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J. Haponiak, S. Breiter and D. Vokrouhlicky
Wed, 1 Apr 20
7/83

Comments: to be published in Celestial Mechanics and Dynamical Astronomy

The Geometry of Isochrone Orbits: from Archimedes' parabolae to Kepler's third law [CL]

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http://arxiv.org/abs/2003.13456


In classical mechanics, the Kepler potential and the Harmonic potential share the following remarkable property: in either of these potentials, a bound test particle orbits with a radial period that is independent of its angular momentum. For this reason, the Kepler and Harmonic potentials are called \it{isochrone}. In this paper, we solve the following general problem: are there any other isochrone potentials, and if so, what kind of orbits do they contain? To answer these questions, we adopt a geometrical point of view initiated by Michel H\’enon in 1959, in order to explore and classify exhaustively the set of isochrone potentials and isochrone orbits. In particular, we provide a geometric generalization of Kepler’s third law, and give a similar law for the apsidal angle, for any isochrone orbit. We also relate the set of isochrone orbits to the set of parabolae in the plane under linear transformations, and use this to derive an analytical parameterization of any isochrone orbit. Along the way we compare our results to known ones, pinpoint some interesting details of this mathematical physics problem, and argue that our geometrical methods can be exported to more generic orbits in potential theory.

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P. Ramond and J. Perez
Tue, 31 Mar 20
57/94

Comments: 54 pages, 18 figures

Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense clusters (ii) Physical properties and negative heat capacity of pre-collapse core [GA]

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http://arxiv.org/abs/2003.13179


This is the second paper of a series of our works on the isotropic self-similar orbit-averaged Fokker-Planck (OAFP) equation and details physical properties of pre-collapse solution. The fundamental core collapse process at the late stage of relaxation evolution of spherical star clusters can be described by the self-similar OAFP equation. The accurate spectral solution was found recently in the first paper. The present work details the thermodynamical aspects of the model based on the stellar DF obtained from the solution. Our calculation shows the following local properties (i) Equation of state in the core is local ideal gas $p=1.0\rho/\chi_\text{esc}$ where the $p$ is the pressure, $\rho$ density and $\chi_\text{esc}$ the scaled escape energy, while it is polytropic $p=0.5\rho^\Gamma/\chi_\text{esc}$ at large radii where $\Gamma$ is the adiabatic index. (ii) If we consider the center is polytropic sphere, the polytropic index is 177. Also, as global property we construct caloric curves of the model to discuss the heat capacity together with Virial. Special focus is the cause of negative heat capacity of the core; the negativity is directly related to the deep potential well or large scaled escape energy through the criterion condition $\phi=-6/\chi_\text{esc}$ where $\phi$ is the central mean field potential in a well-relaxed core. Comparing our results to the previous works, we conclude, in the self-similar evolution, the negative heat capacity in the core holds due to collisionless and high-temperature stars that experience a rapid change in mean field potential through stellar- and heat- flow, rather than the isolation from surroundings due to self-gravity.

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Y. Ito
Tue, 31 Mar 20
79/94

Comments: N/A

Analytic solutions in Einstein-aether scalar field cosmology [CL]

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http://arxiv.org/abs/2003.03903


In the context of Einstein-aether scalar field cosmology we solve the field equations and determine exact and analytic solutions. In particular, we consider a model proposed by Kanno and Soda where the aether and the scalar fields interact through the aether coefficient parameters, which are promoted to be functions of the scalar field. For this model, we write the field equations by using the minisuperspace approach and we determine the scalar field potentials which leads to Liouville–integrable systems. We solve the field equations for five families of scalar field potentials and, whether it is feasible, we write down the analytic solutions by using closed-form functions.

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A. Paliathanasis and G. Leon
Tue, 10 Mar 20
34/63

Comments: 16 pages, 2 figures

Expression of the Holtsmark function in terms of hypergeometric $_2F_2$ and Airy $\mathrm{Bi}$ functions [CL]

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http://arxiv.org/abs/2001.11893


The Holtsmark distribution has applications in plasma physics, for the electric-microfield distribution involved in spectral line shapes for instance, as well as in astrophysics for the distribution of gravitating bodies. It is one of the few examples of a stable distribution for which a closed-form expression of the probability density function is known. However, the latter is not expressible in terms of elementary functions. In the present work, we mention that the Holtsmark probability density function can be expressed in terms of hypergeometric function $_2F_2$ and of Airy function of the second kind $\mathrm{Bi}$ and its derivative. The new formula is simpler than the one proposed by Lee involving $_2F_3$ and $_3F_4$ hypergeometric functions.

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J. Pain
Mon, 3 Feb 20
17/46

Comments: N/A

Second-order Gauge-invariant Cosmological Perturbation Theory: Current Status updated in 2019 [CL]

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http://arxiv.org/abs/1912.12805


The current status of the recent developments of the second-order gauge-invariant cosmological perturbation theory is reviewed. To show the essence of this perturbation theory, we concentrate only on the universe filled with a single scalar field. Through this review, we point out the problems which should be clarified for the further theoretical sophistication of this perturbation theory. This review is an extension of the review paper [K.~Nakamura, “Second-Order Gauge-Invariant Cosmological Perturbation Theory: Current Status”, Advances in Astronomy, vol.2010 (2010), 576273.]. We also expect that this theoretical sophistication will be also useful to discuss the future developments in cosmology as a precise science.

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K. Nakamura
Wed, 1 Jan 20
28/88

Comments: 40 pages, 2 figures, to be published as a book chapter in “Theory and Applications of Physical Science” from Book Publisher International; arXiv admin note: text overlap with arXiv:gr-qc/0605108, arXiv:1001.2621

Second-order Gauge-invariant Cosmological Perturbation Theory: Current Status updated in 2019 [CL]

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http://arxiv.org/abs/1912.12805


The current status of the recent developments of the second-order gauge-invariant cosmological perturbation theory is reviewed. To show the essence of this perturbation theory, we concentrate only on the universe filled with a single scalar field. Through this review, we point out the problems which should be clarified for the further theoretical sophistication of this perturbation theory. This review is an extension of the review paper [K.~Nakamura, “Second-Order Gauge-Invariant Cosmological Perturbation Theory: Current Status”, Advances in Astronomy, vol.2010 (2010), 576273.]. We also expect that this theoretical sophistication will be also useful to discuss the future developments in cosmology as a precise science.

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K. Nakamura
Wed, 1 Jan 20
73/88

Comments: 40 pages, 2 figures, to be published as a book chapter in “Theory and Applications of Physical Science” from Book Publisher International; arXiv admin note: text overlap with arXiv:gr-qc/0605108, arXiv:1001.2621

Study of the Relativistic Dynamics of Extreme-Mass-Ratio Inspirals [CL]

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http://arxiv.org/abs/1912.06584


The principal subject of this thesis is the gravitational two-body problem in the extreme-mass-ratio regime—that is, where one mass is significantly smaller than the other—in the full context of our contemporary theory of gravity, general relativity. We divide this work into two broad parts: the first provides an overview of the theory of general relativity along with the basic mathematical methods underlying it, focusing on its canonical formulation and perturbation techniques; the second presents our novel work in these areas, focusing on the problems of entropy, motion and the self-force in general relativity. We begin here with a study of entropy theorems in classical Hamiltonian systems, and in particular, the issue of the second law of thermodynamics in classical mechanics and general relativity. Then, we develop a general approach based on conservation laws for calculating the correction to the motion of a sufficiently small object due to gravitational perturbations in general relativity. When the perturbations are attributed to the small object itself, this effect is known as the gravitational self-force. It is what drives the orbital evolution of extreme-mass-ratio inspirals: compact binary systems where one mass is much smaller than—thus effectively orbiting and eventually spiralling into—the other, expected to be among the main sources for the future space-based gravitational wave detector LISA. Finally, we present some work on the numerical computation of the scalar self-force using an approach called the Particle-without-Particle method, as well as the generalization of this method to general partial differential equations and applications to other areas of applied mathematics.

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M. Oltean
Mon, 16 Dec 19
42/62

Comments: PhD thesis, Autonomous University of Barcelona and University of Orl\’eans, 2019 (302 pages, 39 figures)

Angular momentum bounds in particle systems [GA]

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http://arxiv.org/abs/1912.01002


Four expressions involving sums of position and velocity coordinates bounding the total angular momentum of particle systems, and by extension of any continuous or discontinuous material systems, are derived which are tighter for any particle configuration than similar inequalities derived by Sundman (1913), Saari (2005) and Scheeres (2012). Eight distinct inequalities can thus be ordered according to their tightness to angular momentum.

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D. Pfenniger
Wed, 4 Dec 19
18/58

Comments: This article published as part of the topical collection on 50 years of Celestial Mechanics and Dynamical Astronomy. The abstract references in the publised version have been altered by the publisher

Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics [CL]

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http://arxiv.org/abs/1911.06295


We study the structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics (SMHD) in the sense of the local-in-time existence and uniqueness of discontinuous solutions satisfying corresponding jump conditions. The equations of SMHD form a symmetric hyperbolic system which is formally analogous to the system of 2D compressible elastodynamics for particular nonphysical deformations. Using this analogy and the recent results in [Morando A., Trakhinin Y., Trebeschi P. Math. Ann. (2019), https://doi.org/10.1007/s00208-019-01920-6] for shock waves in 2D compressible elastodynamics, we prove that shock waves in SMHD are structurally stable if and only if the fluid height increases across the shock front. For current-vortex sheets the fluid height is continuous whereas the tangential components of the velocity and the magnetic field may have a jump. Applying a so-called secondary symmetrization of the symmetric system of SMHD equations, we find a condition sufficient for the structural stability of current-vortex sheets.

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Y. Trakhinin
Fri, 15 Nov 19
37/73

Comments: 18 pages

Extended FLRW Models: dynamical cancellation of cosmological anisotropies [CL]

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http://arxiv.org/abs/1911.05793


We investigate a corner of the Bianchi models that has not received much attention: “extended FLRW models” (eFLRW) defined as a cosmological model with underlying anisotropic Bianchi geometry that nevertheless expands isotropically and can be mapped onto a reference FLRW model with the same expansion history. In order to investigate the stability and naturalness of such models in a dynamical systems context, we consider spatially homogeneous models that contain a massless scalar field $\varphi$ and a non-tilted perfect fluid obeying an equation of state $p=w\rho$. Remarkably, we find that matter anisotropies and geometrical anisotropies tend to cancel out dynamically. Hence, the expansion is asymptotically isotropic under rather general conditions. Although extended FLRW models require a special matter sector with anisotropies that are ‘fine-tuned” relative to geometrical anisotropies, our analysis shows that such solutions are dynamically preferred attractors in general relativity. Specifically, we prove that all locally rotationally symmetric Bianchi type III universes with space-like $\nabla_\mu\varphi$ are asymptotically shear-free, for all $w\in[-1,1]$. Moreover, all shear-free equilibrium sets with anisotropic spatial curvature are proved to be stable with respect to all homogeneous perturbations for $w\geq -1/3$.

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M. Thorsrud, B. Normann and T. Pereira
Fri, 15 Nov 19
60/73

Comments: 31 pages, 2 figures

Stable attractors in the three-dimensional general relativistic Poynting-Robertson effect [CL]

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http://arxiv.org/abs/1911.03649


We prove the stability of the critical hypersurfaces associated with the three-dimensional general relativistic Poynting-Robertson effect. The equatorial ring configures to be as a stable attractor and the whole critical hypersurface as a basin of attraction for this dynamical system. We introduce a new, simpler (in terms of calculations), and more physical approach within the Lyapunov theory. We propose three different Lyapunov functions, each one carrying important information and very useful for understanding such phenomenon under different aspects.

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V. Falco and P. Bakala
Tue, 12 Nov 19
78/84

Comments: 11 pages, 3 figures

Dissipative systems in metric theories of gravity. Foundations and applications of the energy formalism [CL]

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http://arxiv.org/abs/1911.03197


In this paper we introduce a new procedure, termed by us \emph{energy formalism}, to deal with dissipative systems in metric theories of gravity. This approach aims at determining the analytic expression of Rayleigh dissipation function in the context of the inverse problem in the calculus of variations. We describe our method in detail, presenting a simple example. After, we consider as first extensive application the general relativistic Poynting-Robertson effect. The obtained results and future implications are discussed.

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V. Falco and E. Battista
Mon, 11 Nov 19
64/105

Comments: 18 pages, 3 Figures

Asymptotic behavior of a matter filled universe with exotic topology [CL]

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http://arxiv.org/abs/1911.01233


The ADM formalism together with a constant mean curvature (CMC) temporal gauge is used to derive the monotonic decay of a weak Lyapunov function of the Einstein dynamical equations in an expanding universe with a positive cosmological constant and matter sources satisfying suitable energy conditions. While such a Lyapunov function does not, in general, represent a true Hamiltonian of the matter-coupled gravity dynamics (unlike in the vacuum case when it does), it can nevertheless be used to study the asymptotic behavior of the spacetimes. The Lyapunov function attains its infimum only in the limit that the matter sources be turned off or, at least, become asymptotically negligible provided that the universe model does not re-collapse and form singularities. Later we specialize our result to the case of a perfect fluid which satisfies the desired energy conditions. However, even in this special case, we show using Shutz’s velocity potential formalism cast into Hamiltonian form that unlike the vacuum spacetimes (with or without a positive cosmological constant), construction of a true Hamiltonian for the dynamics in constant mean curvature temporal gauge is difficult and therefore the Lyapunov function does not have a straightforward physical interpretation. Nevertheless, we show, for the fluid with equation of state $P=(\gamma-1)\rho$ ($1\leq\gamma\leq2$), that the general results obtained hold true and the infimum of the weak Lyapunov function can be related to the Sigma constant, a topological invariant of the manifold. Utilizing these results, some general conclusions are drawn regarding the asymptotic state of the universe and the dynamical control of the allowed spatial topologies in the cosmological models.

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P. Mondal
Tue, 5 Nov 19
42/72

Comments: N/A

A study of inhomogeneous massless scalar gauge fields in cosmology [CL]

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http://arxiv.org/abs/1909.11965


Why is the Universe so homogeneous and isotropic? We summarize a general study of a $\gamma$-law perfect fluid alongside an inhomogeneous, massless scalar gauge field (with homogeneous gradient) in anisotropic spaces with General Relativity. The anisotropic matter sector is implemented as a $j$-form (field-strength level), where $j\,\in\,{1,3}$, and the spaces studied are Bianchi space-times of solvable type. Wald’s no-hair theorem is extended to include the $j$-form case. We highlight three new self-similar space-times: the Edge, the Rope and Wonderland. The latter solution is so far found to exist in the physical state space of types I,II, IV, VI$_0$, VI$_h$, VII$_0$ and VII$_h$, and is a global attractor in I and V. The stability analysis of the other types has not yet been performed. This paper is a summary of ~[1], with some remarks towards new results which will be further laid out in upcoming work.

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B. Normann, S. Hervik, A. Ricciardone, et. al.
Fri, 27 Sep 19
5/64

Comments: Conference proceedings, 6 pages

Reconstruction Procedure for Nonlocal Gauss-Bonnet Models [CL]

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http://arxiv.org/abs/1909.09452


We investigate the cosmological dynamics of nonlocally corrected gravity involving a function of the inverse d’Alembertian acting on the Gauss-Bonnet term. Casting the dynamical equations in local form, we derive the reconstruction procedure. We find conditions on the model parameters that are sufficient for the existence of de Sitter solutions and obtain these solutions explicitly.

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E. Elizalde, E. Pozdeeva and S. Vernov
Wed, 25 Sep 19
32/70

Comments: 5 pages

Similarity Inner Solutions for the Pulsar Equation [HEAP]

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http://arxiv.org/abs/1909.08521


Lie symmetries are applied to classify the source of the magnetic field for the Pulsar equation near to the surface of the neutron star. We find that there are six possible different admitted Lie algebras. We apply the corresponding Lie invariants to reduce the Pulsar equation close to the surface to an ordinary differential equation. This equation is solved either with the use of Lie symmetries or the application of the ARS algorithm for singularity analysis to write the analytic solution as a Laurent expansion. These solutions are called inner solutions.

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A. Paliathanasis
Thu, 19 Sep 19
23/71

Comments: 11 pages, 2 figures, accepted for publication by Mathematical Methods in the Applied Science

Godbillon-Vey Helicity and Magnetic Helicity in Magnetohydrodynamics [SSA]

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http://arxiv.org/abs/1909.07291


The Godbillon-Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a 3D manifold are satisfied. The magnetic Godbillon-Vey helicity invariant in magnetohydrodynamics (MHD) is a higher order helicity invariant that occurs for flows, in which the magnetic helicity density $h_m={\bf A}{\bf\cdot}{\bf B}={\bf A}{\bf\cdot}(\nabla\times{\bf A})=0$, where ${\bf A}$ is the magnetic vector potential and ${\bf B}$ is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon-Vey field $\boldsymbol{\eta}={\bf A}\times{\bf B}/|{\bf A}|^2$ and the Godbillon-Vey helicity density $h_{gv}=\boldsymbol{\eta}{\bf\cdot}(\nabla\times{\boldsymbol\eta})$ in general MHD flows in which either $h_m=0$ or $h_m\neq 0$. A conservation law for $h_{gv}$ occurs in flows for which $h_m=0$. For $h_m\neq 0$ the evolution equation for $h_{gv}$ contains a source term in which $h_m$ is coupled to $h_{gv}$ via the shear tensor of the background flow. The transport equation for $h_{gv}$ also depends on the electric field potential $\psi$, which is related to the gauge for ${\bf A}$, which takes its simplest form for the advected ${\bf A}$ gauge in which $\psi={\bf A\cdot u}$ where ${\bf u}$ is the fluid velocity. An application of the Godbillon-Vey magnetic helicity to nonlinear force-free magnetic fields used in solar physics is investigated. The possible uses of the Godbillon-Vey helicity in zero helicity flows in ideal fluid mechanics, and in zero helicity Lagrangian kinematics of three-dimensional advection are discussed.

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G. Webb, A. Prasad, S. Anco, et. al.
Tue, 17 Sep 19
22/98

Comments: 42 pages, 10 figures, accepted for publication in Journal of Plasma Physics

Mathematical theory of physical vector fields [HEAP]

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http://arxiv.org/abs/1909.04836


We study the dynamics and statistics of real vector fields in flat (n+1)-dimensional space-time with an emphasis on the field topology and stochasticity in physical applications such as stochastic magnetic and velocity fields in cosmological systems. We show that the natural field topology defined by the metric, induced by the Euclidean vector norm, is physically implausible. However, any vector field corresponds to a dynamical system with a topology in the corresponding phase space. This phase space topology, unlike the natural topology in Euclidean space, is defined using open balls that contain nearby vectors in both vector space and real space and hence is more physical. In addition, it is preserved under time translation if certain conditions including time reversal invariance are satisfied by the field. If these mathematical conditions are not satisfied, therefore, the field’s topology can spontaneously change as the field evolves in time. In this context, similar to topological entropy, which measures the complexity of a dynamical system in the phase space, a simple quantity is defined for a vector field which measures its spatial complexity in real space. For stochastic fields, this spatial complexity can be taken as a measure of the field’s stochasticity level. Generalizing a previous work based on renormalization group invariance, we show that corresponding to any arbitrary vector field, there exists a scalar field whose properties provide a means to quantify the vector field’s spatial complexity, stochasticity level and dissipation rate.

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A. Jafari and E. Vishniac
Thu, 12 Sep 19
62/84

Comments: N/A

Structure of the center manifold of the L1 and L2 collinear libration points in the restricted three-body problem [CL]

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http://arxiv.org/abs/1909.03299


We present a global analysis of the center manifold of the collinear points in the circular restricted three-body problem. The phase-space structure is provided by a family of resonant 2-DOF Hamiltonian normal forms. The near 1:1 commensurability leads to the construction of a detuned Birkhoff-Gustavson normal form. The bifurcation sequences of the main orbit families are investigated by a geometric theory based on the reduction of the symmetries of the normal form, invariant under spatial mirror symmetries and time reversion. This global picture applies to any values of the mass parameter.

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G. Pucacco
Tue, 10 Sep 19
76/80

Comments: Paper included in the Topical Collection for the 50th birthday of CM&DA, 19 pages, 7 figures

Numerical integration in celestial mechanics: a case for contact geometry [CL]

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http://arxiv.org/abs/1909.02613


Several dynamical systems of interest in celestial mechanics can be written in the form
\begin{equation}
\ddot q + \frac{\partial V(q,t)}{\partial q}+f(t)\dot q=0\,. %\quad i=1,\dots,n\,.
\end{equation}
For instance, the modified Kepler problem, the spin–orbit model and the Lane–Emden equation all belong to this class.
In this work we start an investigation of these models from the point of view of contact geometry. In particular we focus on the (contact) Hamiltonisation of these models and on the construction of the corresponding geometric integrators.

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A. Bravetti, M. Seri, M. Vermeeren, et. al.
Mon, 9 Sep 19
60/67

Comments: N/A

Magnetic stochasticity and diffusion [HEAP]

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http://arxiv.org/abs/1908.06474


We develop a quantitative relationship between magnetic diffusion and the level of randomness, or stochasticity, of the diffusing magnetic field in a magnetized medium. A general mathematical formulation of magnetic stochasticity in turbulence has been developed in previous work in terms of the ${\cal L}_p$-norm $S_p(t)={1\over 2}|| 1-\hat{\bf B}_l.\hat{\bf B}_L||_p$, $p$th order magnetic stochasticity of the stochastic field ${\bf B}({\bf x}, t)$, based on the coarse-grained fields, ${\bf B}_l$ and ${\bf B}_L$, at different scales, $l\neq L$. For laminar flows, stochasticity level becomes the level of field self-entanglement or spatial complexity. In this paper, we establish a connection between magnetic stochasticity $S_p(t)$ and magnetic diffusion in magnetohydrodynamic (MHD) turbulence and use a homogeneous, incompressible MHD simulation to test this prediction. Our results agree with the well-known fact that magnetic diffusion in turbulent media follows the super-linear Richardson dispersion scheme. This is intimately related to stochastic magnetic reconnection in which super-linear Richardson diffusion broadens the matter outflow width and accelerates the reconnection process.

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A. Jafari, E. Vishniac and V. Vaikundaraman
Tue, 20 Aug 19
76/86

Comments: N/A

Cylindrically symmetric $n$-dimensional (un)charged de Sitter and anti-de Sitter black holes in generic $f(T)$ gravity [CL]

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http://arxiv.org/abs/1908.04995


Given a generic function $f(T)$ we construct in almost closed forms cylindrically symmetric $n$-dimensional uncharged and charged de Sitter and anti-de Sitter solutions (including black holes, wormholes and possibly other regular solutions) in $f(T)$ gravity. Applications to some known models are considered.

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M. Azreg-Aïnou
Thu, 15 Aug 19
47/69

Comments: 6 pages

Boundary Crossing in Stochastic Inflation with Critical Number of Fields [CL]

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http://arxiv.org/abs/1907.13149


We study boundary crossing probability in the context of stochastic inflation. We prove that for a generic multi-field inflationary potential, the probability that the inflaton reaches infinitely far regions in the field space is critically dependent on the number of fields, being nonzero for more than two fields, and zero otherwise. We also provide several examples where the boundary crossing probability can be calculated exactly, most notably, for a particular landscape of a two-field model with a multi-well potential.

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M. Noorbala and H. Firouzjahi
Thu, 1 Aug 19
60/66

Comments: 22 pages, 7 figures

Static self-gravitating Newtonian elastic balls [CL]

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http://arxiv.org/abs/1907.09970


The existence of static self-gravitating Newtonian elastic balls is proved under general assumptions on the constitutive equations of the elastic material. The proof uses methods from the theory of finite-dimensional dynamical systems and the Euler formulation of elasticity theory for spherically symmetric bodies introduced recently by the authors. Examples of elastic materials covered by the results of this paper are Saint Venant-Kirchhoff, John and Hadamard materials.

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A. Alho and S. Calogero
Wed, 24 Jul 19
19/60

Comments: 29 pages, 2 figures

The Effect Of Cooling On Driven Kink Oscillations Of Coronal Loops [SSA]

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http://arxiv.org/abs/1905.13137


Ever since their detection two decades ago, standing kink oscillations in coronal loops have been extensively studied both observationally and theoretically. Almost all driven coronal loop oscillations (e.g., by flares) are observed to damp through time often with Gaussian or exponential profiles. Intriguingly, however, it has been shown theoretically that the amplitudes of some oscillations could be modified from Gaussian or exponential profiles if cooling is present in the coronal loop systems. Indeed, in some cases the oscillation amplitude can even increase through time. In this article, we analyse a flare-driven coronal loop oscillation observed by the Solar Dynamics Observatory’s Atmospheric Imaging Assembly (SDO/AIA) in order to investigate whether models of cooling can explain the amplitude profile of the oscillation and whether hints of cooling can be found in the intensity evolution of several SDO/AIA filters. During the oscillation of this loop system, the kink mode amplitude appears to differ from a typical Gaussian or exponential profile with some hints being present that the amplitude increases. The application of cooling coronal loop modelling allowed us to estimate the density ratio between the loop and the background plasma, with a ratio of between 2.05-2.35 being returned. Overall, our results indicate that consideration of the thermal evolution of coronal loop systems can allow us to better describe oscillations in these structures and return more accurate estimates of the physical properties of the loops (e.g., density, scale height, magnetic field strength).

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C. Nelson, A. Shukhobodskiy, R. Erdélyi, et. al.
Fri, 31 May 19
36/58

Comments: 7 Figures, 14 pages, Accepted for publication in Frontiers in Astronomy and Space Science

Magnification Cross Sections for the Elliptic Umbilic Caustic Surface [CEA]

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http://arxiv.org/abs/1905.11974


We show that the asymptotic scaling of the magnification volume cross section corresponding to an elliptic umbilic caustic surface is $\mu^{-2.5}$ in the two-image region and $\mu^{-2}$ in the four-image region, where $\mu$ is the total unsigned magnification.

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A. Aazami, C. Keeton and A. Petters
Wed, 29 May 19
13/45

Comments: 5 pages, 5 figures

Accurate modelling of the low-order secondary resonances in the spin-orbit problem [EPA]

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http://arxiv.org/abs/1904.07047


We provide an analytical approximation to the dynamics in each of the three most important low order secondary resonances (1:1, 2:1, and 3:1) bifurcating from the synchronous primary resonance in the gravitational spin-orbit problem. To this end we extend the perturbative approach introduced in Gkolias et. al. (2016), based on normal form series computations. This allows to recover analytically all non-trivial features of the phase space topology and bifurcations associated with these resonances. Applications include the characterization of spin states of irregular planetary satellites or double systems of minor bodies with irregular shapes. The key ingredients of our method are: i) the use of a detuning parameter measuring the distance from the exact resonance, and ii) an efficient scheme to `book-keep’ the series terms, which allows to simultaneously treat all small parameters entering the problem. Explicit formulas are provided for each secondary resonance, yielding i) the time evolution of the spin state, ii) the form of phase portraits, iii) initial conditions and stability for periodic solutions, and iv) bifurcation diagrams associated with the periodic orbits. We give also error estimates of the method, based on analyzing the asymptotic behavior of the remainder of the normal form series.

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I. Gkolias, C. Efthymiopoulos, A. Celletti, et. al.
Tue, 16 Apr 19
67/88

Comments: Accepted for publication in Communications in Nonlinear Science and Numerical Simulation

Normalization of the Levi-Civita Hamiltonian at a collinear Lagrange point [EPA]

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http://arxiv.org/abs/1904.04146


The normalizations of the Hamiltonian of the circular restricted three-body problem at a collinear Lagrange equilibrium are used to compute approximations of its center and tube manifolds, as well as of all the dynamics in their neighbourhood. For small values of the reduced mass m the radius of convergence of any (even partial) normalization at L1,L2 is affected by the complex singularities of the gravitational potential energy of the closest primary body, i.e. the one with smallest mass. In this paper we investigate if regularizations with respect to the body of mass m improve the convergence of the normalizations. In particular, we consider the Hamiltonian describing the planar three-body problem in Levi-Civita regularizing variables, and we show that for a suitable interval of the value of the Hamiltonian larger than the value at the Lagrange equilibrium, the Levi-Civita Hamiltonian has a fictitious center-saddle equilibrium at which the Hamiltonian can be normalized. We find that, for a sample value of m corresponding to the Sun-Jupiter mass ratio, the normalized regularized Hamiltonian provides approximation of the center and of the tube manifolds of L1 up to an energy corresponding to a Lyapunov orbit of amplitude which is larger than the distance |1-m-xL1| of L1 from a complex singularity.

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R. Paez and M. Guzzo
Tue, 9 Apr 19
79/105

Comments: 17 pages, 7 figures. Submitted for publication

Instability of coherent states of a real scalar field [CL]

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http://arxiv.org/abs/0510097


We investigate stability of both localized time-periodic coherent states (pulsons) and uniformly distributed coherent states (oscillating condensate) of a real scalar field satisfying the Klein-Gordon equation with a logarithmic nonlinearity. The linear analysis of time-dependent parts of perturbations leads to the Hill equation with a singular coefficient. To evaluate the characteristic exponent we extend the Lindemann-Stieltjes method, usually applied to the Mathieu and Lame equations, to the case that the periodic coefficient in the general Hill equation is an unbounded function of time. As a result, we derive the formula for the characteristic exponent and calculate the stability-instability chart. Then we analyze the spatial structure of the perturbations. Using these results we show that the pulsons of any amplitudes, remaining well-localized objects, lose their coherence with time. This means that, strictly speaking, all pulsons of the model considered are unstable. Nevertheless, for the nodeless pulsons the rate of the coherence breaking in narrow ranges of amplitudes is found to be very small, so that such pulsons can be long-lived. Further, we use the obtaned stability-instability chart to examine the Affleck-Dine type condensate. We conclude the oscillating condensate can decay into an ensemble of the nodeless pulsons.

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V. Koutvitsky and E. Maslov
Tue, 29 Jan 19
43/62

Comments: 11 pages, 8 figures, submitted to Physical Review E

Causality of the Einstein-Israel-Stewart Theory with Bulk Viscosity [CL]

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http://arxiv.org/abs/1901.06701


We prove that the Einstein-Israel-Stewart equations, describing the dynamics of a relativistic fluid with bulk viscosity and nonzero baryon charge (without shear viscosity or baryon diffusion) dynamically coupled to gravity, are causal in the full nonlinear regime. We also show that these equations can be written as a first-order symmetric hyperbolic system, implying local existence and uniqueness of solutions to the equations of motion. We use an arbitrary equation of state and do not make any simplifying symmetry or near-equilibrium assumption, requiring only physically natural conditions on the fields. These results pave the way for the inclusion of bulk viscosity effects in simulations of gravitational-wave signals coming from neutron star mergers.

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F. Bemfica, M. Disconzi and J. Noronha
Wed, 23 Jan 19
60/111

Comments: 8 pages

Symplectic Coarse-Grained Dynamics: Chalkboard Motion in Classical and Quantum Mechanics [CL]

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http://arxiv.org/abs/1901.06554


In the usual approaches to mechanics (classical or quantum) the primary object of interest is the Hamiltonian, from which one tries to deduce the solutions of the equations of motion (Hamilton or Schr\”odinger). In the present work we reverse this paradigm and view the motions themselves as being the primary objects. This is made possible by studying arbitrary phase space motions, not of points, but of (small) ellipsoids with the requirement that the symplectic capacity of these ellipsoids is preserved. This allows us to guide and control these motions as we like. In the classical case these ellipsoids correspond to a symplectic coarse graining of phase space, and in the quantum case they correspond to the “quantum blobs” we defined in previous work, and which can be viewed as minimum uncertainty phase space cells which are in a one-to-one correspondence with Gaussian pure states.

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M. Gosson
Wed, 23 Jan 19
100/111

Comments: N/A

Chiral effects in magnetized quantum spinor matter in particle and astroparticle physics [CL]

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http://arxiv.org/abs/1812.11099


Quantum spinor matter in extremal conditions (high densities and temperatures, presence of strong magnetic fields) have drawn the attention of researchers in diverse areas of contemporary physics, ranging from cosmology, high-energy and astroparticle physics to condensed matter physics. We study an impact of the confining boundary conditions on the properties of physical systems with hot dense magnetized ultrarelativistic spinor matter and elucidate a significant role of boundaries for such systems.

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Y. Sitenko
Mon, 31 Dec 18
2/57

Comments: 14 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1612.08815, arXiv:1606.08241, arXiv:1603.09268

Comment on "Acceleration of particles to high energy via gravitational repulsion in the Schwarzschild field" by C. H. McGruder III [CL]

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http://arxiv.org/abs/1812.06832


By direct computations, we show that “repulsion” in the Schwarzschild field can not accelerate an outgoing particle, and thus represents pure coordinate effect. In other words, the repulsion can not be detected neither by local nor by distant observer.

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A. Deriglazov, W. Ramírez and P. Rojas
Thu, 20 Dec 18
39/62

Comments: 4 pages

The theory of figures of Clairaut with focus on the gravitational rigidity modulus: inequalities and an improvement in the Darwin-Radau equation [EPA]

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http://arxiv.org/abs/1811.07759


This paper contains a review of Clairaut’s theory with focus on the determination of a gravitational rigidity modulus $\gamma$ defined as $\left(\frac{C-I_o}{I_o}\right)\gamma=\frac{2}{3}\Omega^2$, where $C$ and $I_o$ are the polar and mean moment of inertia of the body and $\Omega$ is the body spin.The constant $\gamma$ is related to the static fluid Love number $k_2= \frac{3I_o G}{R^5} \frac{1}{\gamma}$, where $R$ is the body radius and $G$ is the gravitational constant. The new results are: a variational principle for $\gamma$, upper and lower bounds on the ellipticity that improve previous bounds by Chandrasekhar (1963) and a semi-empirical procedure for estimating $\gamma$ from the knowledge of $m$, $I_o$, and $R$, where $m$ is the mass of the body. The main conclusion is that for $0.2\le I_o/(mR^2)\le 0.4$ the approximation $\gamma\approx G \sqrt{ \frac{2^7}{5^5}\frac{m^5}{I_o^3}}= \gamma_I$ is a better estimate for $\gamma$ than that obtained from the Darwin-Radau equation, denoted as $\gamma_{DR}$. Moreover, within the range of applicability of the Darwin-Radau equation $0.32\le I_o/(mR^2)\le 0.4$ the relative difference between the two estimates, $|\gamma_{DR}/\gamma_I -1|$, is less than $0.05\%$.

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C. Ragazzo
Tue, 20 Nov 18
1/73

Comments: N/A