Curl Component Research Articles

Anisotropic strong lensing as a probe of dark matter self-interactions

ABSTRACT Galaxy-scale strongly lensed systems have been shown to provide a unique technique for exploring the underlying physics of dark matter at sub-galactic scales. In the past, much attention was given to detecting and studying individual haloes in a strong lens system. In addition to the subhaloes, line-of-sight haloes contribute significantly to the small perturbations in lensed images. In prior work, we demonstrated that these line-of-sight haloes imprint a distinctive anisotropic signature and hence give rise to a detectable non-zero parity-even quadrupole moment in the effective convergence field’s two-point correlation function. In this study, we show that these line-of-sight haloes also produce a non-zero curl component of the effective deflection field with a parity-odd quadrupole moment of the two-point function. These multipole moments have the ability to statistically separate line-of-sight haloes from dark matter substructure. In this paper, we examine how these multipole moments evolve in the presence of warm dark matter and self-interacting dark matter in terms of central density evolution and dark matter halo abundance. Importantly, we show that these different multipole moments display exquisite sensitivity to both the amplitude and the velocity dependence of the dark matter self-interaction cross-section. Our approach opens the door for strong lensing observations to probe dark matter self-interaction over a broad range of relative velocities.

Physics-informed neural network for phase imaging based on transport of intensity equation.

Non-interferometric quantitative phase imaging based on Transport of Intensity Equation (TIE) has been widely used in bio-medical imaging. However, analytic TIE phase retrieval is prone to low-spatial frequency noise amplification, which is caused by the illposedness of inversion at the origin of the spectrum. There are also retrieval ambiguities resulting from the lack of sensitivity to the curl component of the Poynting vector occurring with strong absorption. Here, we establish a physics-informed neural network (PINN) to address these issues, by integrating the forward and inverse physics models into a cascaded deep neural network. We demonstrate that the proposed PINN is efficiently trained using a small set of sample data, enabling the conversion of noise-corrupted 2-shot TIE phase retrievals to high quality phase images under partially coherent LED illumination. The efficacy of the proposed approach is demonstrated by both simulation using a standard image database and experiment using human buccal epitehlial cells. In particular, high image quality (SSIM = 0.919) is achieved experimentally using a reduced size of labeled data (140 image pairs). We discuss the robustness of the proposed approach against insufficient training data, and demonstrate that the parallel architecture of PINN is efficient for transfer learning.

Status of Cosmic Microwave Background Observations for the Search of Primordial Gravitational Waves

The cosmic microwave background (CMB) is one of the most powerful tools for cosmology. Its polarization could have imprinted the sign of an inflationary background of gravitational waves, which is supposed to have originated at 10−38/10−35 seconds after the Big Bang. Detecting this background is extremely difficult because of the weakness of the signal (if any) left on the CMB polarization and because of the need to control the systematic effects. Additionally, the presence of astrophysical foregrounds, the possibility of leakage from curl-free to curl-like components, including gravitational lensing, and the instrumental noise and systematics, require sensitive detectors and smart systematic effect control. We discuss the experimental efforts spent in this field, highlighting the key observational difference and the choice that could lead, in the near future, to the detection of the curl component of the CMB polarization, a clear sign of the inflationary expansion.

Urban spatial structures from human flow by Hodge–Kodaira decomposition

Human flow in cities indicates social activity and can reveal urban spatial structures based on human behaviours for relevant applications. Scalar potential is a mathematical concept that, when properly applied, can provide an intuitive view of human flow. However, the definition of such a potential in terms of the origin-destination flow matrix and its feasibility remain unresolved. In this case, we use Hodge–Kodaira decomposition, which uniquely decomposes a matrix into a potential-driven (gradient) flow and a curl flow. We depict the potential landscapes in cities resulting from commuting flow and reveal how the landscapes have either changed or remained unchanged by years or methods of transportation. We then determine how well the commuting flow is described by the potential, by evaluating the percentage of the gradient component for metropolitan areas in the USA and show that the gradient component is almost 100% in several areas; in other areas, however, the curl component is dominant, indicating the importance of circular flow along with triangles of places. The potential landscape provides an easy-to-use visualisation tool for showing the attractive places of human flow and will help in a variety of applications such as commerce, urban design, and epidemic spreading.

Hydrodynamic Properties of the Sun’s Giant Cellular Flows

Abstract Measurements of the large cellular flows on the Sun were made by local correlation tracking of features (supergranules) seen in full-disk Doppler images obtained by the Helioseismic and Magnetic Imager (HMI) instrument on the NASA Solar Dynamics Observatory (SDO) satellite. Several improvements made to the local correlation tracking method allowed for more precise measurements of these flows. Measurements were made hourly over the nearly ten years of the mission-to-date. A four-hour time-lag between images was determined to give the best results as a compromise between increased feature displacement and decreased feature evolution. The hourly measurements were averaged over the 34 days that it takes to observe all longitudes at all latitudes to produce daily maps of the latitudinal and longitudinal velocities. Analyses of these flow maps reveal many interesting characteristics of these large cellular flows. While flows at all latitudes are largely in the form of vortices with left-handed helicity in the north and right-handed helicity in the south, there are key distinctions between the low-latitude and high-latitude cells. The low-latitude cells have roughly circular shapes, lifetimes of about one month, rotate nearly rigidly, do not drift in latitude, and do not exhibit any correlation between longitudinal and latitudinal flow. The high-latitude cells have long extensions that spiral inward toward the poles and can wrap nearly completely around the Sun. They have lifetimes of several months, rotate differentially with latitude, drift poleward at speeds approaching 2 m s−1, and have a strong correlation between prograde and equatorward flows. Spherical harmonic spectral analyses of maps of the divergence and curl of the flows confirm that the flows are dominated by the curl component with rms velocities of about 12 m s−1 at wavenumber ℓ = 10. Fourier transforms in time over 1024 daily records of the spherical harmonic spectra indicate two notable components—an m = ±ℓ feature representing the low-latitude component and an m = ±1 feature representing the high-latitude component. The dispersion relation for the low-latitude component is well represented by that derived for Rossby waves or r-modes. The high-latitude component has a constant temporal frequency for all ℓ indicating features advected by differential rotation at rates representative of the base of the convection zone high latitudes. The poleward motions of these features further suggest that the high-latitude meridional flow at the base of the convection zone is poleward—not equatorward.

Helmholtz–Hodge decomposition-based 2D and 3D ocean surface current visualization for mesoscale eddy detection

Ocean surface current (or ocean flow) visualization plays a significant role in the understanding of dynamical processes of ocean. It has been a hot research topic in both computer science and oceanography. Ocean surface current is a turbulent flow field mixing of multi-scale ocean dynamics such as large-scale ocean circulations ( $$100\,\hbox {km}\sim$$ ), mesoscale eddies (10– $$100\,\hbox {km}$$ ), submesoscale processes (1– $$10\,\hbox {km}$$ ). Mesoscale eddies, which are strong but short-life movement relative to the large-scale ocean circulations, have great importance on the transportation of ocean water masses, momentum and energy. However, their detection and recognition, which are treated as the foundation of exploring their dynamical mechanisms, is still a challenging issue. For one thing, in the mixed ocean flow field, different ocean flows depended and influence with each other making existing methods difficult to identify among them. For another, mesoscale eddies are active signals on the ocean. They may change their forms and velocities at any time. This challenges existing works to deal with their boundary ambiguity and unremitting transitions. To solve these problems, this paper proposes a novel 2D and 3D ocean surface current visualization approach based on an amended Helmholtz–Hodge decomposition (HHD), which can be widely used for mesoscale eddy detection. In our method, HHD decomposes each mixed ocean flow field to two components: curl component and divergence component. Different ocean flows can be represented by these two components independently. In addition, to improve the performance of eddy identification and to reveal the 3D structure of ocean flows simultaneously, HHD transforms the 2D ocean flow field to 3D potential surfaces. Finally, comprehensive experiments are performed on both global and local ocean flow field (Black Sea and Mediterranean Sea) calculated from satellite maps of sea level anomaly to verify our method. Experimental results demonstrate the good effectiveness of our method.

The Quest for B Modes from Inflationary Gravitational Waves

The search for the curl component (B mode) in the cosmic microwave background (CMB) polarization induced by inflationary gravitational waves is described. The canonical single-field slow-roll model of inflation is presented, and we explain the quantum production of primordial density perturbations and gravitational waves. It is shown how these gravitational waves then give rise to polarization in the CMB. We then describe the geometric decomposition of the CMB polarization pattern into a curl-free component (E mode) and curl component (B mode) and show explicitly that gravitational waves induce B modes. We discuss the B modes induced by gravitational lensing and by Galactic foregrounds and show how both are distinguished from those induced by inflationary gravitational waves. Issues involved in the experimental pursuit of these B modes are described, and we summarize some of the strategies being pursued. We close with a brief discussion of some other avenues toward detecting/characterizing the inflationary gravitational-wave background.

Internal energy flows of coma-affected singular beams in low-numerical-aperture systems.

The circulating phase gradient component of a singular beam gets modified when focused by a low-numerical-aperature system suffering from coma aberration. The gradient due to this coma aberration splits the higher charge vortex into elementary vortices and distributes them spatially. This splitting depends on the charge and polarity of the incident singular beam as well as the sign and magnitude of the aberration coefficient. The transverse component of the Poynting vector field distribution at the focal plane is decomposed into the curl or solenoidal component and divergence or irrotational component using the Helmholtz-Hodge decomposition technique. The solenoidal component that relates to the orbital angular momentum carries the circulating energy, while the irrotational component shows the sources and sinks of the energy. Intriguing results of the study of energy flow around the edge dislocations apart from the point phase defects in the irrotational components are also presented.

Rotation of the Cosmic Microwave Background Polarization from Weak Gravitational Lensing

When a cosmic microwave background (CMB) photon travels from the surface of last scatter through spacetime metric perturbations, the polarization vector may rotate about its direction of propagation. This gravitational rotation is distinct from, and occurs in addition to, the lensing deflection of the photon trajectory. This rotation can be sourced by linear vector or tensor metric perturbations and is fully coherent with the curl deflection field. Therefore, lensing corrections to the CMB polarization power spectra as well as the temperature-polarization cross correlations due to nonscalar perturbations are modified. The rotation does not affect lensing by linear scalar perturbations, but needs to be included when calculations go to higher orders. We present complete results for weak lensing of the full-sky CMB power spectra by general linear metric perturbations, taking into account both deflection of the photon trajectory and rotation of the polarization. For the case of lensing by gravitational waves, we show that the B modes induced by the rotation largely cancel those induced by the curl component of deflection.

Peculiar velocity decomposition, redshift space distortion, and velocity reconstruction in redshift surveys: The methodology

Massive spectroscopic surveys will measure the redshift space distortion (RSD) induced by galaxy peculiar velocity to unprecedented accuracy and open a new era of precision RSD cosmology. We develop a new method to improve the RSD modeling and to carry out robust reconstruction of the 3D large scale peculiar velocity through galaxy redshift surveys, in light of RSD. (1) We propose a mathematically unique and physically motivated decomposition of peculiar velocity into three eigencomponents: an irrotational component completely correlated with the underlying density field (${\mathbf{v}}_{\ensuremath{\delta}}$), an irrotational component uncorrelated with the density field (${\mathbf{v}}_{S}$), and a rotational (curl) component (${\mathbf{v}}_{B}$). The three components have different origins, different scale dependences, and different impacts on RSD. (2) This decomposition has the potential to simplify and improve the RSD modeling. (i) ${\mathbf{v}}_{B}$ damps the redshift space clustering. (ii) ${\mathbf{v}}_{S}$ causes both damping and enhancement to the redshift space power spectrum ${P}^{s}(k,u)$. Nevertheless, the leading order contribution to the enhancement has a ${u}^{4}$ directional dependence, distinctively different from the Kaiser formula. Here, $u\ensuremath{\equiv}{k}_{z}/k$, $k$ is the amplitude of the wave vector, and ${k}_{z}$ is the component along the line of sight. (iii) ${\mathbf{v}}_{\ensuremath{\delta}}$ is of the greatest importance for the RSD cosmology. We find that the induced redshift clustering shows a number of important deviations from the usual Kaiser formula. Even in the limit of ${\mathbf{v}}_{S}\ensuremath{\rightarrow}0$ and ${\mathbf{v}}_{B}\ensuremath{\rightarrow}0$, the leading order contribution $\ensuremath{\propto}(1+f\stackrel{\texttildelow{}}{W}(k){u}^{2}{)}^{2}$. It differs from the Kaiser formula by a window function $\stackrel{\texttildelow{}}{W}(k)$. Nonlinear evolution generically drives $\stackrel{\texttildelow{}}{W}(k)\ensuremath{\le}1$. We hence identify a significant systematical error causing underestimation of the structure growth parameter $f$ by as much as $O(10%)$ even at a relatively large scale $k=0.1h/\mathrm{Mpc}$. (iv) The velocity decomposition reveals the three origins of the ``finger-of-God'' (FOG) effect and suggests how to simplify and improve the modeling of FOG by treating the three components separately. (v) We derive a new formula for the redshift space power spectrum. Under the velocity decomposition scheme, all high order Gaussian corrections and non-Gaussian corrections of order ${\ensuremath{\delta}}^{3}$ can be taken into account without introducing extra model uncertainties. Here $\ensuremath{\delta}$ is the nonlinear overdensity. (3) The velocity decomposition clarifies issues in peculiar velocity reconstruction through 3D galaxy distribution. We discuss two possible ways to carry out the 3D ${\mathbf{v}}_{\ensuremath{\delta}}$ reconstruction. Both use the otherwise troublesome RSD in velocity reconstruction as a valuable source of information. Both have the advantage of rendering the reconstruction of a stochastic 3D field into the reconstruction of a deterministic window function ${W}^{s}(k,u)$ of limited degrees of freedom. Both can automatically and significantly alleviate the galaxy bias problem and, in the limit of a deterministic galaxy bias, completely overcome it. Part 1 of this series of works lays out the methodology. Companion papers Y. Zheng et al. (in preparation). will extensively evaluate its performance against N-body simulations.

Mixed-motion Segmentation Using Helmholtz Decomposition

Motion segmentation plays a central role in video analysis, such as the surveillance, human-computer interaction, action recognition, etc. Extensive studies have been done on the stationary camera scenarios. Recently, more attentions are focusing on dynamic backgrounds with several moving objects in the scene. In many applications, the background motion is of much less interest, and solely the local object motion is expected. Several approaches have been proposed for global motion estimation and motion segmentation (the global motion and the object motion are also named as inlier and outlier, respectively). The work in [6] introduced a parametric form which assumed the global motion model from simple translation to general perspective transformation using different parameters. A joint global motion estimation and segmentation method was proposed in [2]. It iteratively updates the inlier model by segmenting the outlier out. A regression scheme, using gradient descent (GD) [6] or least squares (LS) [5], is also applied to refine the inlier model. An outlier rejection filter in [1] explicit filters motion vectors by checking their similarity in a pre-defined window. RANSAC in [3] is a statistical method which estimates the inlier model by iteratively updating the probability of inlier. All above methods are 2D based methods. They require the multiple motions to be independent. But for interdependent motions, they may fail to deal with. Considering this, it is better to place different motions on different layers in higher dimensional space. To this end, our method transforms 2D motion field into 3D surfaces. Local extremes on the surface such as peaks, ridges and valleys depict local motions while smoothing places represent the global motion. The 3D surface is calculated using Helmoholtz decomposition. In [4], a particle filter based on Helmholtz decomposition was proposed for flow estimation. Following is its fundamental theory. For an arbitrary flow field ξ, it is decomposed into two components: curl-free (divergence) component ∇E and divergence-free (curl) component ∇ × W , where E and W are what we want to obtain the 3D potential surfaces. The

Feature-aware streamline generation of planar vector fields via topological methods

Methods for vector field visualization strive to have a sparse representation of the field, while encoding all of its important features. Streamline visualization is one of the most popular such methods. Traditionally, a set of streamline methods have focused on capturing the salient features of the vector field such as sources, sinks, and vortices. However, not all features are created equal, and some features of the vector field are more important than others, which could simply be mere noise. It is this problem of characterizing feature importance through streamline visualization that we try to address in this paper. Specifically, a given 2D vector field can be decomposed into a rotation-free (gradient) component, divergence-free (curl) component and a harmonic component by the so-called Hodge decomposition. Features in the original vector field, in some sense, correspond to features in the first two components. Furthermore, the gradient and curl components are each induced by a scalar field. By analyzing these two corresponding scalar fields using topological methods (in particular the contour tree and the persistent homology), we develop a simple and novel algorithm whose streamline density tends to reflect the topological importance of the features in the input vector field. Such a feature-aware streamline sketch is more informative, yet still simple both visually and in terms of its generation. It enhances our understanding of the underlying vector field, which is demonstrated here by several experimental results.

Lensing of 21-cm Fluctuations by Primordial Gravitational Waves

Weak-gravitational-lensing distortions to the intensity pattern of 21-cm radiation from the dark ages can be decomposed geometrically into curl and curl-free components. Lensing by primordial gravitational waves induces a curl component, while the contribution from lensing by density fluctuations is strongly suppressed. Angular fluctuations in the 21-cm background extend to very small angular scales, and measurements at different frequencies probe different shells in redshift space. There is thus a huge trove of information with which to reconstruct the curl component of the lensing field, allowing tensor-to-scalar ratios conceivably as small as r~10(-9)-far smaller than those currently accessible-to be probed.

Power asymmetries in the cosmic microwave background temperature and polarization patterns

We test the asymmetry of the Cosmic Microwave Background anisotropy jointly in temperature and polarization. We study the hemispherical asymmetry, previously found only in the temperature field, with respect to the axis identified by Hansen et al. (2009). To this extent, we make use of the low resolution WMAP 5 year temperature and polarization Nside=16 maps and the optimal angular power spectrum estimator BolPol (Gruppuso et al. 2009). We consider two simple estimators for the power asymmetry and we compare our findings with Monte Carlo simulations which take into account the full noise covariance matrix. We confirm an excess of power in temperature angular power spectrum in the Southern hemisphere at a significant level, between 3 sigma and 4 sigma depending on the exact range of multipoles considered. We do not find significant power asymmetry in the gradient (curl) component EE (BB) of polarized angular spectra. Also cross-correlation power spectra, i.e. TE, TB, EB, show no significant hemispherical asymmetry. We also show that the Cold Spot found by Vielva et al. (2004) in the Southern Galactic hemisphere does not alter the significance of the hemispherical asymmetries on multipoles which can be probed by maps at resolution Nside=16. Although the origin of the hemispherical asymmetry in temperature remains unclear, the study of the polarization patter could add useful information on its explanation. We therefore forecast by Monte Carlo the Planck capabilities in probing polarization asymmetries.

Spatial variation of the fine-structure parameter and the cosmic microwave background

We study the effects on cosmic microwave background (CMB) temperature and polarization anisotropies of spatial fluctuations of the fine-structure parameter between causally disconnected regions of the Universe at the time of recombination. Analogous to weak gravitational lensing, in addition to modifying the mean power spectra and inducing a curl component $(B$ mode) to the polarization, spatial fluctuations of the fine-structure parameter induce higher-order (non-Gaussian) temperature and polarization correlations in the CMB. We calculate these effects for the general case of arbitrary correlation between temperature fluctuations and fine-structure parameter fluctuations, and show the results for a model where these two types of fluctuations are uncorrelated. The formalism we present here may also be applied to other modifications of recombination physics that do not significantly alter the evolution of the dominant density perturbations. We discuss the constraints on the effective Lagrangian for the variable fine-structure parameter necessary to realize this scenario.

Reconstruction of lensing from the cosmic microwave background polarization

Gravitational lensing of the cosmic microwave background (CMB) polarization field has been recognized as a potentially valuable probe of the cosmological density field. We apply likelihood-based techniques to the problem of lensing of CMB polarization and show that if the B-mode polarization is mapped, then likelihood-based techniques allow significantly better lensing reconstruction than is possible using the previous quadratic estimator approach. With this method the ultimate limit to lensing reconstruction is not set by the lensed CMB power spectrum. Second-order corrections are known to produce a curl component of the lensing deflection field that cannot be described by a potential; we show that this does not significantly affect the reconstruction at noise levels greater than $0.25\ensuremath{\mu}\mathrm{K}$ arcmin. The reduction of the mean squared error in the lensing reconstruction relative to the quadratic method can be as much as a factor of two at noise levels of $1.4\ensuremath{\mu}\mathrm{K}$ arcmin to a factor of ten at $0.25\ensuremath{\mu}\mathrm{K}$ arcmin, depending on the angular scale of interest.

The intensity potential approach

Sound intensity vectors can be decomposed into irrotational and curl components. The irrotational part describes intensity propagation, and the curl component vortices and near field of sources. To predict power flow one can limit the problem to irrotational intensity, that is to say to the determination of intensity potential. The resulting equation is analogous to heat transfer, permitting one to use standard heat transfer solvers to predict acoustic power flow. The intensity potential approach is presented for acoustic propagation from sources in cavities with apertures. The modelization of absorbing materials through thermal convection factor is discussed. Finally, comparison with exact prediction and experimental results are presented.

Separation of gravitational-wave and cosmic-shear contributions to cosmic microwave background polarization.

Inflationary gravitational waves (GW) contribute to the curl component in the polarization of the cosmic microwave background (CMB). Cosmic shear--gravitational lensing of the CMB--converts a fraction of the dominant gradient polarization to the curl component. Higher-order correlations can be used to map the cosmic shear and subtract this contribution to the curl. Arcminute resolution will be required to pursue GW amplitudes smaller than those accessible by the Planck surveyor mission. The blurring by lensing of small-scale CMB power leads with this reconstruction technique to a minimum detectable GW amplitude corresponding to an inflation energy near 10(15) GeV.

Discriminating Weak Lensing from Intrinsic Spin Correlations Using the Curl‐Gradient Decomposition

The distortion field defined by the ellipticities of galaxy shapes projected on the sky can be uniquely decomposed into a gradient and a curl component. If the observed ellipticities are induced by weak gravitational lensing, then the distortion field is curl free. Here we show that, in contrast, the distortion field resulting from intrinsic spin alignments is not curl free. This provides a powerful discriminant between lensing and intrinsic contributions to observed ellipticity correlations. We also show how these contributions can be disentangled statistically from the ellipticity correlations or computed locally from circular integrals of the ellipticity field. This allows for an unambiguous detection of intrinsic galaxy alignments in the data. When the distortions are dominated by lensing, as occurs at high redshifts, the decomposition provides a valuable tool for understanding properties of the noise and systematic errors. These techniques can be applied equally well to the polarization of the microwave background, where it can be used to separate curl-free scalar perturbations from those produced by gravity waves or defects.

Polarization pursuers’ guide

We calculate the detectability of the polarization of the cosmic microwave background (CMB) as a function of the sky coverage, angular resolution, and instrumental sensitivity for a hypothetical experiment. We consider the gradient component of the polarization from density perturbations (scalar modes) and the curl component from gravitational waves (tensor modes). We show that the amplitude (and thus the detectability) of the polarization from density perturbations is roughly the same in any model as long as the model fits the big-bang-nucleosynthesis (BBN) baryon density and degree-scale anisotropy measurements. The degree-scale polarization is smaller (and accordingly more difficult to detect) if the baryon density is higher. We show that the sensitivity to the polarization from density perturbations and gravitational waves is improved (by a factor of 30) in a fixed-time experiment with a deeper survey of a smaller region of sky.