Equilateral Configuration Research Articles

Tensor non-Gaussianity in chiral scalar-tensor theories of gravity

Violation of parity symmetry in the gravitational sector, which manifests into unequal left and right circular polarization states of primordial gravitational waves, represents a way to test high-energy modifications to general relativity. In this paper we study inflation within recently proposed chiral scalar-tensor theories of gravity, that extend Chern-Simons gravity by including parity-violating operators containing first and second derivatives of the non-minimally coupled scalar (inflaton) field. Given the degeneracy between different parity-violating theories at the level of the power spectrum statistics, we make a detailed analysis of the parity violation on primordial tensor non-Gaussianity. We show, with an explicit computation, that no new contributions arise in the graviton bispectra if the couplings in the new operators are constant in a pure de Sitter phase. On the other hand, if the coupling functions are time-dependent during inflation, the tensor bispectra acquire non-vanishing contributions from the parity-breaking operators even in the exact de Sitter limit, with maximal signal in the squeezed and equilateral configurations. We also comment on the consistency relation of the three-point function of tensor modes in this class of models and discuss prospects of detecting parity-breaking signatures through Cosmic Microwave Background B-mode bispectra.

Manifold Optimization for High-Accuracy Spatial Location Estimation Using Ultrasound Waves

This paper designs a high accuracy spatial location estimation method using ultrasound waves by exploiting the fixed geometry of the transmitters. Assuming an equilateral triangle antenna configuration, where three antennas are placed as the vertices of an equilateral triangle, the spatial location problem can be formulated as a non-convex optimization problem whose interior is shown to admit a Riemannian manifold structure. The investigation of the geometry of the newly introduced manifold, i.e. the manifold of all equilateral triangles in R^3, allows the design of highly efficient optimization algorithms. Simulation results are presented to compare the performance of the proposed approach against popular methods from the literature. The results suggest that the proposed Riemannian-based methods outperform the state-of-the-art methods. Furthermore, the proposed Riemannian methods require much smaller computation time as compared with popular generic non-convex approaches.

Five-body central configurations and symmetry

We study five-body central configurations. The three-body equilateral triangle Lagrange’s central configuration and four-body equilateral tetrahedron Lehmann-Filhès’s central configuration were generalized to an equilateral five-body central configuration in four dimensions. Our working tool is a coordinate system that had many useful properties in considering central configurations of three, four, and five bodies. From this equilateral solution, projecting to three dimensions, we compute concave and convex five-body central configurations of five different positive masses. We discover the possibility of geometric symmetry of these solutions when two of the masses are equal. Then we study the plane symmetric central configurations with a kite figure. This figure has three masses on the symmetry axis of the kite and two equal masses symmetrically placed on a transversal orthogonal line to the axis. Our parameters are now the masses and the distances along the axis of symmetry. The distances are measured with respect to the crossing point of the diagonals. In this paper, we find and solve a linear equation for these three masses in terms of the distances. We will draw an example of typical elliptic trajectories obeying the Newton equations of motion in such a kite central configuration.

Improved reconstruction of a stochastic gravitational wave background with LISA

We present a data analysis methodology for a model-independent reconstruction of the spectral shape of a stochastic gravitational wave background with LISA. We improve a previously proposed reconstruction algorithm that relied on a single Time-Delay-Interferometry (TDI) channel by including a complete set of TDI channels. As in the earlier work, we assume an idealized equilateral configuration. We test the improved algorithm with a number of case studies, including reconstruction in the presence of two different astrophysical foreground signals. We find that including additional channels helps in different ways: it reduces the uncertainties on the reconstruction; it makes the global likelihood maximization less prone to falling into local extrema; and it efficiently breaks degeneracies between the signal and the instrumental noise.

Forecasts on primordial non-Gaussianity from 21 cm intensity mapping experiments

We forecast the ability of dedicated 21 cm intensity mapping experiments to constraint primordial non-Gaussianity using power spectrum and bispectrum. We model the signal by including the non-linear biasing expansion through a generalized halo model approach. We consider the importance of foreground filtering scale and of the foreground wedge. We find that the current generation intensity mapping experiments like CHIME do not posses sufficient sensitivity to be competitive with the existing limits. On the other hand, upcoming experiments like HIRAX can improve the current constraints and the proposed PUMA experiment can substantially improve them, reaching sensitivities below σ(fNL)<5 for equilateral and orthogonal configurations and σ(fNL) < 1 for the local shape, if good foreground control is achieved.

The effect of radiation pressure on the basins of convergence in the restricted four-body problem

Abstract The present problem deals with the effect of the radiation pressure on the topology of the basins of convergence in the restricted problem of four bodies in two different configurations: (I) the equilateral four-body configuration (II) the collinear four-body problem. We have illustrated the basins of convergence linked to the in-plane as well as out-of-plane libration points by applying the Newton-Raphson multivariate iterative scheme, in both the configurations. Additionally, the basin entropy is illustrated to estimate the degree of fractality of the basins of convergence.

He8 and Li9 cluster structures in light nuclei

The possibility of the $^8$He and $^{9}$Li clusters in atomic nuclei is discussed. Until now most of the clusters in the conventional models have been limited to the closures of the three-dimensional harmonic oscillators, such as $^4$He, $^{16}$O, and $^{40}$Ca. In the neutron-rich nuclei, however, the neutron to proton ratio is not unity, and it is worthwhile to think about more neutron-rich objects with $N>Z$ as the building blocks of cluster structures. Here the nuclei with the neutron number six, which is the subclosure of the $p_{3/2}$ subshell of the $jj$-coupling shell model, are assumed to be clusters, and thus we study the $^8$He and $^9$Li cluster structures in $^{16}$Be ($^8$He+$^8$He), $^{17}$B ($^8$He+$^9$Li), $^{18}$C ($^9$Li+$^9$Li), and $^{24}$C ($^8$He+$^8$He+$^8$He). Recent progress of the antisymmetrized quasi cluster model (AQCM) enables us to utilize $jj$-coupling shell model wave functions as the clusters rather easily. It is shown that the $^8$He+$^9$Li and $^9$Li+$^9$Li cluster configurations cover the lowest shell-model states of $^{17}$B and $^{18}$C, respectively. To predict the cluster states with large relative distances, we increase the expectation value of the principal quantum numbers by adding the nodes to the lowest states under the condition that the total angular momentum is unchanged (equal to $J^\pi =0$). As a result, developed cluster states are obtained around the corresponding threshold energies. The rotational band structure of $^{24}$C, which reflect the symmetry of equilateral triangular configuration ($D_{3h}$ symmetry) of three $^8$He clusters, also appears around the threshold energy.

The near wake of three circular cylinders in an equilateral triangular arrangement at a low Reynolds number Re=100

Two-dimensional numerical simulations are conducted at a low Reynolds number Re = 100 to investigate the near wake of three identical circular cylinders that are arranged in an equilateral triangular configuration. The incident angle of the three-cylinder configuration with respect to incoming flow is varied from θ= 0° to 60°, while the spacing between adjacent cylinders (L) covers a wide range of L/D = 1.25-7.0, where D is diameter of the cylinder. Typical flow structures in the near wake of the three-cylinder configuration are identified, including a single Karman vortex street, bistable flip-flopping near wake, anti-phase and/or in-phase vortex shedding, shear layer reattachment, and vortex impingement, depending on the configuration (L/D, θ). The behavior of Strouhal number (St) is discussed in detail, echoing the distinct structures of near wake. Furthermore, fluid forces on the individual cylinders are examined, which, though highly depending on (L/D, θ), exhibit a close correlation to the near wake behavior.

Modelling the matter bispectrum at small scales in modified gravity

Future large-scale structure surveys will measure three-point statistics with high statistical significance. This will offer significant improvements on our understanding of gravity, provided we can model these statistics accurately. {We assess the performance of several schemes for theoretical modelling of the matter bispectrum, including halo-model based approaches and fitting formulae. We compare the model predictions against N-body simulations, considering scales up to kmax = 4 h/Mpc, well into non-linear regime of structure formation. Focusing on the equilateral configuration, we conduct this analysis for three theories of gravity: general relativity, f(R) gravity, and the DGP braneworld model. Additionally, we compute the lensing convergence bispectrum for these models. We find that all current modelling prescriptions in modified gravity, in particular for theories with scale-dependent linear growth, fail to attain the accuracy required by the precision of the Stage IV surveys such as Euclid. Among these models, we find that a halo-model corrected fitting formula achieves the best overall performance.}

On then-body problem in R4

Using geometric mechanics methods, we examine aspects of the dynamics ofnmass points inR4with a general pairwise potential. We investigate the central force problem, set up then-body problem and discuss certain properties of relative equilibria. We describe regularn-gons inR4, and when the masses are equal we determine the invariant manifold of motions with regularn-gon configurations. In the casen = 3, we reduce the dynamics to a 6 d.f. system and we show that for generic potentials and momenta, relative equilibria with equilateral configuration are unstable.This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

Experimental Study of Heat Exchange and Aerodynamic and Thermal Contact Resistance of Staggered Bundles of Tubes with Spiral Rolled Fins of Air-Cooled Devices

Results of a study of the thermal and dynamic characteristics of three types of six-row checkerboard bundles in a transverse flow in the interval of Reynolds number Re = (3−18) × 103 are presented. The experimental data are generalized by similarity equations. The heat-exchange coefficients of each transverse row of the bundle are measured and features that arise in the variation of heat exchange by row of the bundle are identified. The mean value of the thermal contact resistance in the range of variation of the temperature of the contact surfaces 79−95°С is found to be equal to 2.13·10− 4 m2·K/W. On the basis of results of the study it is shown that the use of an equilateral configuration of tubes in bundles should be rejected and replaced by an isosceles configuration with steps S1 = 68−69 mm and S2 = 55.4 mm, a configuration that is advantageous in terms of energy and on the basis of a cost-benefit analysis.

CMB lensing bispectrum: Assessing analytical predictions against full-sky lensing simulations

Cosmic microwave background (CMB) lensing is an integrated effect whose kernel is greater than half the peak value in the range $1<z<5$. Measuring this effect offers a powerful tool to probe the large-scale structure of the Universe at high redshifts. With the increasing precision of ongoing CMB surveys, other statistics than the lensing power spectrum, in particular the lensing bi-spectrum, will be measured at high statistical significance. This will provide ways to improve the constraints on cosmological models and lift degeneracies. Following on an earlier paper, we test analytical predictions of the CMB lensing bi-spectrum against full-sky lensing simulations, and discuss their validity and limitation in detail. The tree-level prediction of perturbation theory agrees with the simulation only up to $\ell\sim 200$, but the one-loop order allows capturing the simulation results up to $\ell\sim 600$. We also show that analytical predictions based on fitting formulas for the matter bi-spectrum agree reasonably well with simulation results, although the precision of the agreement depends on the configurations and scales considered. For instance, the agreement is at the $10\%$-level for the equilateral configuration at multipoles up to $\ell\sim2000$, but the difference in the squeezed limit raises to more than a factor of two at $\ell\sim2000$. This discrepancy appears to come from limitations in the fitting formula of the matter bi-spectrum. We also find that the analytical prediction for the post-Born correction to the bi-spectrum is in good agreement with the simulation. We conclude by discussing the bi-spectrum prediction in some theories of modified gravity.

Tree-level bispectrum in the effective field theory of large-scale structure extended to massive neutrinos

We compute the tree-level bispectrum of dark matter in the presence of massive neutrinos in the mildly non-linear regime in the context of the effective field theory of large-scale structure (EFTofLSS) . For neutrinos, whose typical free streaming wavenumber (kfs) is longer than the non-linear scale (kNL), we solve a Boltzmann equation coupled to the effective fluid equation for dark matter. We solve perturbatively the coupled system by expanding in powers of the neutrino density fraction (fν) and the ratio of the wavenumber of interest over the non-linear scale (k/kNL) and add suitable counterterms to remove the dependence from short distance physics. For equilateral configurations, we find that the total-matter tree-level bispectrum is approximately 16fν times the dark matter one on short scales (k > kfs). The largest contribution stems from the back-reaction of massive neutrinos on the dark matter growth factor. On large scales (k < kfs) the contribution of neutrinos to the bispectrum is smaller by up to two orders of magnitude.

Hilbert’s ‘Monkey Saddle’ and Other Curiosities in the Equilibrium Problem of Three Point Particles on a Circle for Repulsive Power Law Forces

This article determines all possible (proper as well as pseudo) equilibrium arrangements under a repulsive power law force of three point particles on the unit circle. These are the critical points of the sum over the three (standardized) Riesz pair interaction terms, each given by $$V_s(r)= s^{-1}\left( r^{-s}-1 \right) $$ when the real parameter $$s \ne 0$$, and by $$V_0(r) := \lim _{s\rightarrow 0}V_s(r) = -\ln r$$; here, r is the chordal distance between the particles in the pair. The bifurcation diagram which exhibits all these equilibrium arrangements together as functions of s features three obvious “universal” equilibria, which do not depend on s, and two not-so-obvious continuous families of s-dependent non-universal isosceles triangular equilibria. The two continuous families of non-universal equilibria are disconnected, yet they bifurcate off of a common universal limiting equilibrium (the equilateral triangular configuration), at $$s=-4$$, where the graph of the total Riesz energy of the 3-particle configurations has the shape of a “monkey saddle.” In addition, one of the families of non-universal equilibria also bifurcates off of another universal equilibrium (the antipodal arrangement), at $$s=-2$$. While the bifurcation at $$s=-4$$ is analytical, the one at $$s=-2$$ is not. The bifurcation analysis presented here is intended to serve as template for the treatment of similar N-point equilibrium problems on $${\mathbb {S}}^d$$ for small N.

Non-Gaussianity of inflationary gravitational waves from the field equation

We demonstrate equivalence of the in–in formalism and Green’s function method for calculating the bispectrum of primordial gravitational waves generated by vacuum fluctuations of the metric. The tree-level bispectrum from the field equation, [Formula: see text], agrees with the results obtained previously using the in–in formalism exactly. Characterizing non-Gaussianity of the fluctuations using the ratio [Formula: see text] in the equilateral configuration, where [Formula: see text] is the power spectrum of scale-invariant gravitational waves, we show that it is much weaker than in models with spectator gauge fields. We also calculate the tree-level bispectrum of two right-handed and one left-handed gravitational wave using Green’s function, reproducing the results from in–in formalism, and show that it can be as large as the bispectrum of three right-handed gravitational waves.

Gridded and direct Epoch of Reionisation bispectrum estimates using the Murchison Widefield Array

AbstractWe apply two methods to estimate the 21-cm bispectrum from data taken within the Epoch of Reionisation (EoR) project of the Murchison Widefield Array (MWA). Using data acquired with the Phase II compact array allows a direct bispectrum estimate to be undertaken on the multiple redundantly spaced triangles of antenna tiles, as well as an estimate based on data gridded to the uv-plane. The direct and gridded bispectrum estimators are applied to 21 h of high-band (167–197 MHz; z = 6.2–7.5) data from the 2016 and 2017 observing seasons. Analytic predictions for the bispectrum bias and variance for point-source foregrounds are derived. We compare the output of these approaches, the foreground contribution to the signal, and future prospects for measuring the bispectra with redundant and non-redundant arrays. We find that some triangle configurations yield bispectrum estimates that are consistent with the expected noise level after 10 h, while equilateral configurations are strongly foreground-dominated. Careful choice of triangle configurations may be made to reduce foreground bias that hinders power spectrum estimators, and the 21-cm bispectrum may be accessible in less time than the 21-cm power spectrum for some wave modes, with detections in hundreds of hours.

W-state preparation and entanglement dynamics in Rydberg atomic system based on the collective excitation enhancement

We present a theoretical scheme to realize the highly entangled W state in the neutral atomic system by using the Rydberg blockade effect. Considering three neutral atoms trapped in an optical lattice with an equilateral triangle geometric configuration, we show that only one atom can be excited to the Rydberg excite state due to the Rydberg blockade effect. By choosing a non-zero detuning from the exact two-photon resonance in order to compensate for the blockade error caused by the finite interaction energy, we show that the three atoms W state can be generated. Moreover, we show that three atoms are entangled when W state is achieved with high fidelity.

Flattened non-Gaussianities from the effective field theory of inflation with imaginary speed of sound

Inflationary perturbations in multi-field theories can exhibit a transient tachyonic instability as a consequence of their non-trivial motion in the internal field space. When an effective single-field description is applicable, the resulting theory is characterized by fluctuations that propagate with an imaginary speed of sound. We use the effective field theory of fluctuations to study such a set-up in a model-independent manner, highlighting the peculiarities and subtleties that make it different from the standard case. In particular, perturbations feature exponentially growing and decaying modes whose relative amplitude is undetermined within the effective field theory. Nevertheless, we prove that in an interesting limit the dimensionless bispectrum is in fact universal, depending only on the speed of sound and on the cutoff scale that limits the validity of the effective theory. Contrary to the power spectrum, we find that the bispectrum does not display an exponential enhancement. The amplitude of non-Gaussianities in the equilateral configuration is similar to the one of conventional models, but it is enhanced in flattened configurations in a way that is ultraviolet sensitive.

Numerical evaluation of inflationary 3-point functions on curved field space—with the transport method & CppTransport

We extend the public CPPTRANSPORT code to calculate the statistical properties of fluctuations in multiple-field inflationary models with curved field space. Our implementation accounts for all physical effects at tree-level in the `in-in' diagrammatic expansion. This includes particle production due to time-varying masses, but excludes scenarios where the curvature perturbation is generated by averaging over the decay of more than one particle. We test our implementation by comparing results in Cartesian and polar field-space coordinates, showing excellent numerical agreement and only minor degradation in compute time. We compare our results with the PYTRANSPORT 2.0 code, which uses the same computational approach but a different numerical implementation, finding good agreement. Finally, we use our tools to study a class of gelaton-like models which could produce an enhanced non-Gaussian signal on equilateral configurations of the Fourier bispectrum. We show this is difficult to achieve using hyperbolic field-space manifolds and simple inflationary potentials.

Lowering the self-coupling of the scalar field in the generalized Higgs inflation

We study cosmological dynamics of a generalized Higgs inflation. By expanding the action up to the second and third order in the small perturbations, we study the primordial perturbation and its non-Gaussian distribution. We study the non-Gaussian feature in both the equilateral and orthogonal configurations. By adopting a quartic potential, we perform a numerical analysis on the model's parameter space and compare the results with Planck2015 observational data. To obtain some observational constraint, we focus on the self-coupling and the non-minimal coupling parameters. We show that, in the presence of the non-minimal coupling and the Galileon-like interaction, the self-coupling parameter can be reduced to the order of $10^{-6}$ which is much larger than the value that CMB normalization suggests for this self-coupling.