Academic literature on the topic 'Tensor Minkowski Functionals'

A fundamental understanding of the formation and properties of a complex spatial structure relies on robust quantitative tools to characterize morphology. A systematic approach to the characterization of average properties of anisotropic complex interfacial geometries is provided by integral geometry which furnishes a family of morphological descriptors known as tensorial Minkowski functionals. These functionals are curvature-weighted integrals of tensor products of position vectors and surface normal vectors over the interfacial surface. We here demonstrate their use by application to non-cubic triply periodic minimal surface model geometries, whose Weierstrass parametrizations allow for accurate numerical computation of the Minkowski tensors.

Abstract The Minkowski tensors (MTs) can be used to probe anisotropic signals in a field, and are well suited for measuring the redshift-space distortion (RSD) signal in large-scale structure catalogs. We consider how the linear RSD signal can be extracted from a field without resorting to the plane-parallel approximation. A spherically redshift-space distorted field is both anisotropic and inhomogeneous. We derive expressions for the two-point correlation functions that elucidate the inhomogeneity, and then explain how the breakdown of homogeneity impacts the volume and ensemble averages of the tensor Minkowski functionals. We construct the ensemble average of these quantities in curvilinear coordinates and show that the ensemble and volume averages can be approximately equated, but this depends on our choice of definition of the volume average of a tensor and the radial distance between the observer and field. We then extract the tensor Minkowski functionals from spherically redshift-space distorted, Gaussian random fields and gravitationally evolved dark matter density fields at z = 0 to test if we can successfully measure the Kaiser RSD signal. For the dark matter field, we find a significant, ∼10% anomalous signal in the MT component parallel to the line of sight that is present even on large scales R G ≳ 15 Mpc, in addition to the Kaiser effect. This is due to the line-of-sight component of the MT being significantly contaminated by the Finger of God effect, which can be approximately modeled by an additional damping term in the cumulants.

Geometrical and statistical properties of polarization of CMB are analyzed. Singular points of the vector field which describes CMB polarization are found and classified. Statistical distribution of the singularities is studied. A possible signature of tensor perturbations in CMB polarization is discussed. For a further analysis of CMB statistics, Minkowski functionals are used, which present a technically simple method to search for deviations from a Gaussian distribution.

Aims. We develop an automatic bubble-recognition routine based on Minkowski functionals (MF) and tensors (MT) to detect bubble-like interstellar structures in optical emission line images. Methods. Minkowski functionals and MT are powerful mathematical tools for parameterizing the shapes of bodies. Using the papaya2-library, we created maps of the desired MF or MT of structures at a given window size. We used maps of the irreducible MT ψ2, which is sensitive to elongation, to find filamentary regions in Hα, [S II], and [O III] images of the Magellanic Cloud Emission Line Survey. Using the phase of ψ2, we were able to draw lines perpendicular to each filament and thus obtain line-density maps. This allowed us to find the center of a bubble-like structure and to detect structures at different window sizes. Results. The detected bubbles in all bands are spatially correlated to the distribution of massive stars, showing that we indeed detect interstellar bubbles without large spatial bias. Eighteen out of 59 supernova remnants in the Large Magellanic Cloud (LMC) and 13 out of 20 superbubbles are detected in at least one wavelength. The lack of detection is mostly due to surrounding emission that disturbs the detection, a too small size, or the lack of a (circular) counterpart in our emission line images. In line-density maps at larger scales, maxima can be found in regions with high star formation in the past, often inside supergiant shells (SGS). In SGS LMC 2, there is a maximum west of the shell where a collision of large gas clouds is thought to have occurred. In the Small Magellanic Cloud (SMC), bubble detection is impaired by the more complex projected structure of the galaxy. Line maps at large scales show large filaments in the SMC in a north-south direction, especially in the [S II] image. The origin of these filaments is unknown.

Integration over connection type variables in the path integral for the discrete form of the first-order formulation of general relativity theory is studied. The result (a generalized function of the rest of variables of the type of tetrad or elementary areas) can be defined through its moments, i.e. integrals of it with the area tensor monomials. In our previous paper these moments have been defined by deforming integration contours in the complex plane as if we had passed to a Euclidean-like region. In this paper we define and evaluate the moments in the genuine Minkowski region. The distribution of interest resulting from these moments in this non-positively defined region contains the divergences. We prove that the latter contribute only to the singular (δ-function like) part of this distribution with support in the non-physical region of the complex plane of area tensors while in the physical region this distribution (usual function) confirms that defined in our previous paper which decays exponentially at large areas. Besides that, we evaluate the basic integrals over which the integral over connections in the general path integral can be expanded.

I develop the idea that our world is a brane-like object embedded in Euclidean bulk. In its ground state, the brane constituent matter is assumed to be homogeneous and isotropic, and of negligible influence on the bulk geometry. The analysis of this paper is model independent, in the sense that action functional of bulk fields is not specified. Instead, the behavior of the brane is derived from the universally valid conservation equation of the bulk stress tensor. The present work studies the behavior of a 3-sphere in the five-dimensional Euclidean bulk. The sphere is made of bulk matter characterized by the equation of state [Formula: see text]. It is shown that stability of brane vibrations requires [Formula: see text]. Then, the stable brane perturbations obey Klein–Gordon-like equation with an effective metric of Minkowski signature. The argument is given that it is this effective metric that is detected in physical measurements. The corresponding effective Universe is analyzed for all the values of [Formula: see text]. In particular, the effective metric is shown to be a solution of Einstein’s equations coupled to an effective perfect fluid. As an illustration, one simple choice of the brane constituent matter is studied in detail.

Abstract The bright polarized synchrotron emission, away from the Galactic plane, originates mostly from filamentary structures. We implement a filament finder algorithm which allows the detection of bright elongated structures in polarized intensity maps. We analyse the sky at 23 and 30 GHz as observed respectively by WMAP and Planck. We identify 19 filaments, 13 of which have been previously observed. For each filament, we study the polarization fraction, finding values typically larger than for the areas outside the filaments, excluding the Galactic plane, and a fraction of about 30% is reached in two filaments. We study the polarization spectral indices of the filaments, and find a spectral index consistent with the values found in previous analysis (about -3.1) for more diffuse regions. Decomposing the polarization signals into the E and B families, we find that most of the filaments are detected in PE , but not in PB . We then focus on understanding the statistical properties of the diffuse regions of the synchrotron emission at 23 GHz. Using Minkowski functionals and tensors, we analyse the non-Gaussianity and statistical isotropy of the polarized intensity maps. For a sky coverage corresponding to 80% of the fainter emission, and on scales smaller than 6 degrees (ℓ > 30), the deviations from Gaussianity and isotropy are significantly higher than 3σ. The level of deviation decreases for smaller scales, however, it remains significantly high for the lowest analised scale (∼ 1.5°). When 60% sky coverage is analysed, we find that the deviations never exceed 3σ. Finally, we present a simple data-driven model to generate non-Gaussian and anisotropic simulations of the synchrotron polarized emission. The simulations are fitted in order to match the spectral and statistical properties of the faintest 80% sky coverage of the data maps.

Cosmic Microwave Background (CMB) is a relic from the early Universe. It was generated due to the physical processes in the early Universe during an epoch known as the recombination or decoupling epoch. The CMB has highly uniform temperature over the entire sky but with small variations in different directions. The Thomson scattering between photons and electrons during the decoupling epoch, which contain quadrupole anisotropies results in the linear polarization of the CMB. The CMB radiation along each line of sight is associated with temperature (T) and polarization. The polarization can be decomposed into Stokes parameters Q=U, or E mode (E) and B mode (B) fields. Here, Q=U fields transform as spin +_2 objects under rotation transformation while the E=B fields remain invariant. The fluctuations observed in CMB is due to the quantum fluctuations generated during the inflationary phase, which is a period of exponential expansion moments after the Big Bang in the early Universe. The statistical properties of CMB fluctuations will be similar to the primordial fluctuations for the linear evolution of fluctuations. The statistical observables are used to capture the morphological properties of the CMB fluctuations. Then the morphological properties can be studied in relation to the parameters describing the physical mechanisms of the inflationary phase. In this research work, we use the geometrical and topological observables to study the CMB polarization fields and further we also introduce a novel statistical observable known as the Tensor Minkowski Functionals (TMFs) for the analysis of CMB fields. The models about the inflationary phase predict that the Probability Distribution Function (PDF) of primordial fluctuations are close to the Gaussian distribution with small deviation. The information about the exact form of deviation in the PDF of primordial fluctuation is encapsulated in the CMB Fields. We investigate the local type non-Gaussian features in the CMB polarization fields, which is parametrized by fNL. We use the simulations of local type non-Gaussian CMB fields, namely T, E and IP, and study the deviation in their PDF relative to the Gaussian distribution. The numerical calculations show that the non-Gaussian deviation in the PDF of E field is similar to that of the T field. While the non-Gaussian deviation corresponding to the IP field has smaller amplitude and large error bars in comparison to that of T field. This analysis was repeated using the geometrical and topological observables, namely Scalar Minkowski Functionals (SMFs) and Betti numbers of fields. These observables capture different morphological features of a given field. The results obtained using these observables are similar to those from the PDF of the Fields. Hence from the theoretical point of view, these results imply that the E field can provide an independent constraint on fNL similar to the T field. Further, the results show that when the IP field is independently used for such analysis, it cannot provide any statistically significant information. In the realistic scenario, the observational data contains instrumental systematics which will lead to the reduction in the statistical significance of the above results. The CMB polarization is usually analyzed using the E=B fields as they are scalar fields. We investigate the theoretical aspects of using the Q=U fields as a complementary analysis of CMB polarization. We show that the variance of Q=U and its gradient Fields are invariant under rotation transformation; hence it follows that the SMFs of a Gaussian Q=U fields is invariant. However, this statement breaks down for incomplete sky. Then we studied the non-Gaussian deviation in Q=U fields constructed from the simulations of local type non-Gaussian E field. These simulations use the same x􀀀y coordinates along each line of sight. We found that its amplitude is about an order of magnitude lower than that of T Filed and has different shape. This finding will be useful for distinguishing different non-Gaussian signals in the observational data from future experiments. Further, we studied the effect of the presence of primordial tensor perturbation, which is parametrized by r, on the SMFs of Q=U and IP fields, and the number density of singularities in IP Filed. Here, a singularity is a point on the CMB field where IP = 0. We found that the amplitude of SMFs of these Fields are sensitive to the presence of primordial tensor perturbation and it decreases with r. We also show that the number density of singularities in IP Filed decreases with r. This finding will be useful for the searches of primordial tensor perturbation in the future experiments. The instrumental systematics in the observational data will decrease the statistical significance of the above results. TMFs are tensor generalization of Minkowski Functionals which we introduce a new statistical observable for the analysis of CMB data. Since these are tensor quantities, they are capable of capturing more morphological properties in a given Filed than their scalar counterparts. We have developed a code, referred as TMF Code, to compute the TMFs for any general Filed on a Euclidean plane. In order to apply the TMFs, specifically W1;1 2 which is a tensor of rank 2, to CMB Fields which lies on a 2-d spherical surface, we map each point on the sphere with a point on a plane using stereo-graphic projection. The code computes W1;1 2, and then the net anisotropy (_) and net orientation (_) of the structures are estimated. We investigated the numerical error in this computation due to pixelization. We found the error in _ increases with the increasing curvature of the boundaries of the structure. The error in _ is negligible when the structures are completely unoriented with each other and it increases as the structures become more and more aligned with each other. We present the numerical calculation of the systematic variation of A and B with the threshold value for the simulated Gaussian and isotropic CMB T and E Fields. We found that the value of _ shows a characteristic variation with the threshold value while _ is at. We show that according to the standard model, _ = 0:62 for T and _ = 0:63 for E, where the values are corrected for pixelization. The value of _ is one for both the Fields, which is as expected for an isotropic Filed. We applied W1;1 2 for the analysis of PLANCK data as an illustration of its application. The instrumental systematics and the gravitational lensing due to large scale structure affects the morphological features of the CMB Fields. We study the effect of these factors on the value of A and Busing the simulations of CMB frequency bands, namely 44GHz and 70GHz provided in PLANCK data, which contains the respective instrumental characteristics. We found that the percentage difference in A and B due to these factors are less than 2% and it significantly increases the size of their error bars. We use the CMB simulations corresponding to the frequency band 44GHz as the basis for testing the consistency of different PLANCK data sets with theoretical expectations. We estimated the deviation in A and B for the foreground cleaned CMB maps namely SMICA, COMMANDER, SEVEM and NILC corresponding to full mission, half mission 1, half mission 2, half ring 1 and half ring 2 provided in the PLANCK data. These calculations showed that B is consistent with the standard model within 2_ for all data sets, except the T map of NILC half mission 2 which has slightly higher deviation. We found the values of _ for T map of different data sets to be in excellent agreement with the standard model within 1:2􀀀_. The deviation in _ of E map of all data sets are higher than 3_ except the SMICA full mission data. Further, _ for E map corresponding to the half mission 1 of all data sets showed consistently higher deviation of 5 These results imply that the structures in the E map has an extent of alignment with each other. This alignment could be cosmological or due to instrumental systematics. Since we are comparing the PLANCK maps which are obtained by co-adding all frequency bands with that of the simulations with the instrumental characteristics of a specific frequency band, namely 44GHz, the instrumental systematics is more probable reason for the alignment measured in E map.

This chapter proves the main theorems. It begins with global regularity and asymptotics. The chapter considers the future-directed causal geodesics in spacetime, and proves that they extend forever (in the affine parametrization) and become asymptotically parallel to the geodesics of the Minkowski space. It then turns to weak peeling estimates for the Riemann tensor of spacetime. The rates of decay of the components of the Riemann tensor are mainly determined by their signatures. The chapter shows that weak peeling estimates are invariant under natural changes of frames. Meanwhile, the ADM energy measures the total deviation of spacetime from the Minkowski solution. The chapter also explores gauge conditions and parameterizations, before looking at the construction of Bondi energy functions, with good monotonicity properties along null infinity. In order to get precise information on the asymptotic behavior of the metric in the physical space, one needs to understand the bending of the light cones caused by the long-range effect of the nonlinearity (i.e., the modified scattering).