Bipolar Spherical Harmonics Research Articles

Capturing Statistical Isotropy Violation with Generalized Isotropic Angular Correlation Functions of Cosmic Microwave Background Anisotropy

The exquisitely measured maps of fluctuations in the cosmic microwave background (CMB) present the possibility of systematically testing the principle of statistical isotropy of the Universe. A systematic approach based on strong mathematical formulation allows any nonstatistical isotropic (nSI) feature to be traced to the nature of physical effects or observational artifacts. Bipolar spherical harmonics (BipoSH) representation has emerged as an overarching general formalism for quantifying the departures from statistical isotropy for a field on a 2D sphere. We adopt a little-known reduction of the BipoSH functions, dubbed minimal harmonics in the original paper by Manakov et al. We demonstrate that this reduction technique of BipoSH leads to a new generalized set of isotropic angular correlation functions referred to here as minimal BipoSH functions that are observable quantifications of nSI features in a sky map. This paper presents a novel observable quantification of deviation from statistical isotropy in terms of generalized angular correlation functions that are compact and complementary to the BipoSH spectra that generalize the angular power spectrum of CMB fluctuations.

Numerical evaluation of time-distance helioseismic sensitivity kernels in spherical geometry

Context. Helioseismic analysis of large-scale flows and structural inhomogeneities in the Sun requires the computation of sensitivity kernels that account for the spherical geometry of the Sun, as well as systematic effects such as line-of-sight projection. Aims. I aim to develop a code to evaluate helioseismic sensitivity kernels for flows using line-of-sight projected measurements. Methods. I decomposed the velocity field in a basis of vector spherical harmonics and computed the kernel components corresponding to the coefficients of velocity in this basis. The kernels thus computed are radial functions that set up a 1.5D inverse problem to infer the flow from surface measurements. I demonstrate that using the angular momentum addition formalism lets us express the angular dependence of the kernels as bipolar spherical harmonics, which may be evaluated accurately and efficiently. Results. Kernels for line-of-sight projected measurements may differ significantly from those that don’t account for projection. Including projection in our analysis does not increase the computational time significantly. We demonstrate that it is possible to evaluate kernels for pairs of points that are related through a rotation by linearly transforming the terms that enter the expression of the kernel, and that this result holds even for line-of-sight projected kernels. Conclusions. I developed a Julia code that may be used to evaluate sensitivity kernels for seismic wave travel times computed using line-of-sight projected measurements, which is made freely available under the MIT license.

Bayesian estimation of our local motion from the Planck-2018 CMB temperature map

The largest fluctuation in the CMB sky is the CMB dipole, which is believed to be caused by the motion of our observation frame with respect to the CMB rest frame. This motion accounts for the known motion of the Solar System barycentre with a best-fit amplitude of 369 km/s, in the direction (ℓ= 264°, b=48°) in galactic coordinates. Along with the CMB dipole signal, this motion also causes an inevitable signature of statistical anisotropy in the higher multipoles due to the modulation and aberration of the CMB temperature and polarization fields. This leads to a correlation between adjacent CMB multipoles causing a non-zero value of the off-diagonal terms in the covariance matrix which can be captured in terms of the dipolar spectra of the bipolar spherical harmonics (BipoSH). In our work, we jointly infer the CMB power spectrum and the BipoSH spectrum in a Bayesian framework using the Planck-2018 SMICA temperature map. We detect amplitude and direction of the local motion consistent with the canonical value v=369 km/s inferred from CMB dipole with a statistical significance of 4.54σ, 4.97σ and 5.23σ respectively from the masked temperature map with the available sky fraction 40.1%, 59.1%, and 72.2%, confirming the common origin of both the signals. The Bayes factor in favor of the canonical value is between 7 to 8 depending on the choice of mask. But it strongly disagrees (by a value of the Bayes factor about 10-10–10-11) with a higher value of local motion which one can infer from the amplitude of the dipole signal obtained from the CatWISE2020 quasar catalog using the WISE and NEOWISE data set.

Singular value decomposition of the longitudinal ray transform of vector fields in a ball in cone beam coordinates

An analytical singular value decomposition (SVD) of the longitudinal cone-beam transform of solenoidal vector fields in the ball is proposed and described in detail. We exploit the solenoidal vector fields , and constructed by Derevtsov, Kazantsev and Schuster in (2007 J. Inverse Ill-Posed Problems 15 173–185). Our study technique also corresponds to the work Kazantsev (2015 J. Inverse Ill-Posed Problems 23 173–185) in which the scalar case was considered. The calculations use expansions in orthogonal bipolar spherical harmonics. Also the spherical convolution operator Hilbert type , coupling integrals of vector spherical harmonics and Clebsch–Gordan coefficients are involved in our study. The exact formulas for the corresponding singular values , and are obtained and their asymptotic for n → ∞ is studied.

The search for statistical anisotropy in the gravitational-wave background with pulsar timing arrays

Pulsar-timing arrays (PTAs) are seeking gravitational waves from supermassive-black-hole binaries, and there are prospects to complement these searches with stellar-astrometry measurements. Theorists still disagree, however, as to whether the local gravitational-wave background will be statistically isotropic, as arises if it is the summed contributions from many SMBH binaries, or whether it exhibits the type of statistical anisotropy that arises if the local background is dominated by a handful (or even one) bright source. Here we derive, using bipolar spherical harmonics, the optimal PTA estimators for statistical anisotropy in the GW background and simple estimates of the detectability of this anisotropy. We provide results on the smallest detectable amplitude of a dipole anisotropy (and several other low-order multipole moments) and also the smallest detectable amplitude of a "beam'' of gravitational waves. Results are presented as a function of the signal-to-noise with which the GW signal is detected and as a function of the number of pulsars (assuming uniform distribution on the sky and equal sensitivity per pulsar). We provide results first for measurements with a single time-domain window function and then show how the results are augmented with the inclusion of time-domain information. The approach here is intended to be conceptually straightforward and to complement the results of more detailed (but correspondingly less intuitive) modeling of the actual measurements.

Joint Bayesian analysis of large angular scale CMB temperature anomalies

Cosmic microwave background measurements show an agreement with the concordance cosmology model except for a few notable anomalies: Power Suppression, the lack of large scale power in the temperature data compared to what is expected in the concordance model, and Cosmic Hemispherical Asymmetry, a dipolar breakdown of statistical isotropy. An expansion of the CMB covariance in Bipolar Spherical Harmonics naturally parametrizes both these large-scale anomalies, allowing us to perform an exhaustive, fully Bayesian joint analysis of the power spectrum and violations of statistical isotropy up to the dipole level. Our analysis sheds light on the scale dependence of the Cosmic Hemispherical Asymmetry. Assuming a scale-dependent dipole modulation model with a two-parameter power law form, we explore the posterior pdf of amplitude A(l = 16) and the power law index α and find the maximum a posteriori values A*(l = 16) = 0.064 ± 0.022 and α* = −0.92 ± 0.22. The maximum a posteriori direction associated with the Cosmic Hemispherical Asymmetry is (l,b) = (247.8o, −19.6o) in Galactic coordinates, consistent with previous analyses. We evaluate the Bayes factor BSI−DM to compare the Cosmic Hemispherical Asymmetry model with the isotropic model. The data prefer but do not substantially favor the anisotropic model (BSI−DM=0.4). We consider several priors and find that this evidence ratio is robust to prior choice. The large-scale power suppression does not soften when jointly inferring both the isotropic power spectrum and the parameters of the asymmetric model, indicating no evidence that these anomalies are coupled.

A fuzzy bipolar celestial sphere

We introduce a non-commutative deformation of the algebra of bipolar spherical harmonics supporting the action of the full Lorentz algebra. Our construction is close in spirit to the one of the non-commutative spherical harmonics associated to the fuzzy sphere and, as such, it leads to a maximal value of the angular momentum. We derive the action of Lorentz boost generators on such non-commutative spherical harmonics and show that it is compatible with the existence of a maximal angular momentum.

Redshift-space distortions from vector perturbations. II. Anisotropic signal

We study the impact on the galaxy correlation function of the presence of a vector component in the tracers' peculiar velocities, in the case in which statistical isotropy is violated. We present a general framework - based on the bipolar spherical harmonics expansion - to study this effect in a model independent way, without any hypothesis on the origin or the properties of these vector modes. We construct six new observables, that can be directly measured in galaxy catalogs in addition to the standard monopole, quadrupole and hexadecapole, and we show that they completely describe any deviations from isotropy. We then perform a Fisher analysis in order to quantify the constraining power of future galaxy surveys. As an example, we show that the SKA2 would be able to detect anisotropic rotational velocities with amplitudes as low as 1% of that of the vorticity generated during shell-crossing in standard dark matter scenarios.

Reconstruction of a direction-dependent primordial power spectrum from Planck CMB data

We consider the possibility that the primordial curvature perturbation is direction-dependent. To first order this is parameterised by a quadrupolar modulation of the power spectrum and results in statistical anisotropy of the CMB, which can be quantified using `bipolar spherical harmonics'. We compute these for the Planck DR2-2015 SMICA map and estimate the noise covariance from Planck Full Focal Plane 9 simulations. A constant quadrupolar modulation is detected with 2.2 σ significance, dropping to 2σ when the primordial power is assumed to scale with wave number k as a power law. Going beyond previous work we now allow the spectrum to have arbitrary scale-dependence. Our non-parametric reconstruction then suggests several spectral features, the most prominent at k ∼ 0.006 Mpc−1. When a constant quadrupolar modulation is fitted to data in the range 0.005 ⩽ k/Mpc−1 ⩽ 0.008, its preferred directions are found to be related to the cosmic hemispherical asymmetry and the CMB dipole. To determine the significance we apply two test statistics to our reconstructions of the quadrupolar modulation from data, against reconstructions of realisations of noise only. With a test statistic sensitive only to the amplitude of the modulation, the reconstructions from the multipole range 30 ⩽ ℓ ⩽ 1200 are unusual with 2.1σ significance. With the second test statistic, sensitive also to the direction, the significance rises to 6.9σ. Our approach is easily generalised to include other data sets such as polarisation, large-scale structure and forthcoming 21-cm line observations which will enable these anomalies to be investigated further.

Potential energy and dipole moment surfaces for HF@C60: Prediction of spectral and electric response properties.

We present a five-dimensional potential energy surface (PES) for the HF@C60 system computed at the DF-LMP2/cc-pVTZ level of theory. We also calculated a five-dimensional dipole moment surface (DMS) based on DFT(PBE0)/cc-pVTZ calculations. The HF and C60 molecules are considered rigid with bond length rHF = 0.9255 Å (gas phase ground rovibrational state geometry). The C60 geometry is of Ih symmetry. The ab initio points were fitted to obtain a PES in terms of bipolar spherical harmonics. The minimum of the PES corresponds to a geometry where the center of mass of HF is located 0.11 Å away from the center of the cage with an interaction energy of -6.929 kcal/mol. The DMS was also represented in terms of bipolar spherical harmonics. The PES was used to calculate the rotation-translation bound states of HF@C60, and good agreement was found relative to the available experimental data [A. Krachmalnicoff et al., Nat. Chem. 8, 953 (2016)] except for the splitting of the first rotational excitation levels. We propose an empirical adjustment to the PES in order to account for the experimentally observed symmetry breaking. The form of that effective PES is additive. We also propose an effective Hamiltonian with an adjusted rotational constant in order to quantitatively reproduce the experimental results including the splitting of the first rotational state. We use our models to compute the molecular volume polarizability of HF confined by C60 and obtain good agreement with experiment.

Limits on statistical anisotropy from BOSS DR12 galaxies using bipolar spherical harmonics

We measure statistically anisotropic signatures imprinted in three-dimensional galaxy clustering using bipolar spherical harmonics (BipoSHs) in both Fourier space and configuration space. We then constrain a well-known quadrupolar anisotropy parameter $g_{2M}$ in the primordial power spectrum, parametrized by $P(\vec{k}) = \bar{P}(k) [ 1 + \sum_{M} g_{2M} Y_{2M}(\hat{k}) ]$, with $M$ determining the direction of the anisotropy. Such an anisotropic signal is easily contaminated by artificial asymmetries due to specific survey geometry. We precisely estimate the contaminated signal and finally subtract it from the data. Using the galaxy samples obtained by the Baryon Oscillation Spectroscopic Survey Data Release 12, we find no evidence for violation of statistical isotropy, $g_{2M}$ for all $M$ to be of zero within the $2\sigma$ level. The $g_{2M}$-type anisotropy can originate from the primordial curvature power spectrum involving a directional-dependent modulation $g_* (\hat{k} \cdot \hat{p})^2$. The bound on $g_{2M}$ is translated into $g_*$ as $-0.09 < g_* < 0.08$ with a $95\%$ confidence level when $\hat{p}$ is marginalized over.

The Wigner–Eckart Theorem for Reducible Symmetric Cartesian Tensor Operators

We explicitly establish a unitary correspondence between spherical irreducible tensor operators and cartesian tensor operators of any rank. That unitary relation is implemented by means of a basis of integer-spin wave functions that constitute simultaneously a basis of the spaces of cartesian and spherical irreducible tensors. As a consequence, we extend the Wigner--Eckart theorem to cartesian irreducible tensor operators of any rank, and to totally symmetric reducible ones. We also discuss the tensorial structure of several standard spherical irreducible tensors such as ordinary, bipolar and tensor spherical harmonics, spin-polarization operators and multipole operators. As an application, we obtain an explicit expression for the derivatives of any order of spherical harmonics in terms of tensor spherical harmonics.

New geometric representations of the CMB two-point correlation function

When searching for deviations of statistical isotropy in CMB, a popular strategy is to write the two-point correlation function (2pcf) as the most general function of four spherical angles (i.e., two unit vectors) in the celestial sphere. Then, using a basis of bipolar spherical harmonics, statistical anisotropy will show up if and only if any coefficient of the expansion with non-trivial bipolar momentum is detected -- although this detection will not in general elucidate the origin of the anisotropy. In this work we show that two new sets of four angles and basis functions exist which completely specifies the 2pcf, while, at the same time, offering a clearer geometrical interpretation of the mechanisms generating the signal. Since the coefficients of these expansions are zero if and only if isotropy holds, they act as a simple and geometrically motivated null test of statistical isotropy, with the advantage of allowing cosmic variance to be controlled in a systematic way. We report the results of the application of these null tests to the latest temperature data released by the Planck collaboration.

Orthogonal bipolar spherical harmonics measures: Scrutinizing sources of isotropy violation

The two point correlation function of the CMB temperature anisotropies is generally assumed to be statistically isotropic (SI). Deviations from this assumption could be traced to physical or observational artefacts and systematic effects. Measurement of non-vanishing power in the BipoSH spectra is a standard statistical technique to search for isotropy violations. Although this is a neat tool allowing a blind search for SI violations in the CMB sky, it is not easy to discern the cause of isotropy violation using this measure. In this article, we propose a novel technique of constructing orthogonal BipoSH estimators, which can be used to discern between models of isotropy violation.

Statistically anisotropic Gaussian simulations of the CMB temperature field

Although theoretically expected to be Statistically Isotropic (SI), the observed Cosmic Microwave Background (CMB) temperature \& polarization field would exhibit SI violation due to various inevitable effects like weak lensing, Doppler boost and practical limitations of observations like non-circular beam, masking etc. However, presence of any SI violation beyond these effects may lead to a discovery of inherent cosmic SI violation in the CMB temperature \& polarization field. Recently, Planck presented strong evidence of SI violation as a dipolar power asymmetry of the CMB temperature field in two hemispheres. Statistical studies of SI violation effect require non-SI (nSI) Gaussian realizations of CMB temperature field. The nSI Gaussian temperature field leads to non-zero off-diagonal terms in the Spherical Harmonics (SH) space covariance matrix encoded in the coefficients of the Bipolar Spherical Harmonics (BipoSH) representation. We discuss an effective numerical algorithm, Code for Non-Isotropic Gaussian Sky (CoNIGS) to generate nSI realizations of Gaussian CMB temperature field of Planck like resolution with specific cases of SI violation. Realizations of nSI CMB temperature field are obtained for non-zero quadrupolar $(L=2)$ BipoSH measurements by WMAP, dipolar asymmetry (resembles $L=1$ BipoSH coefficients) with a \emph{scale dependent} modulation field as measured by Planck and for Doppler boosted CMB temperature field which also leads to $L=1$ BipoSH spectra. Our method, CoNIGS can incorporate any kind of SI violation and can produce nSI realizations efficiently.

Seeking inflation fossils in the cosmic microwave background

If during inflation the inflaton couples to a ``fossil'' field---some new scalar, vector, or tensor field---it typically induces a scalar-scalar-fossil bispectrum. Even if the fossil field leaves no direct physical trace after inflation, it gives rise to correlations between different Fourier modes of the curvature or, equivalently, a nonzero curvature trispectrum, but without a curvature bispectrum. Here we quantify the effects of a fossil field on the cosmic microwave background temperature fluctuations in terms of bipolar spherical harmonics (BiPoSHs). The effects of vector and tensor fossils can be distinguished geometrically from those of scalars through the parity of the BiPoSHs they induce. However, the two-dimensional nature of the cosmic microwave background sky does not allow vectors to be distinguished geometrically from tensors. We estimate the detectability of a signal in terms of the scalar-scalar-fossil coupling for scalar, vector, and tensor fossils, assuming a local-type coupling. We comment on a divergence that arises in the quadrupolar BiPoSH from the scalar-scalar-tensor correlation in single-field slow-roll inflation.

Exploring above-threshold ionization of hydrogen in an intense x-ray laser field through nonperturbative calculations

This paper addresses the problem of above-threshold ionization (ATI) of hydrogen interacting with a short and intense x-ray electromagnetic field. We consider the weakly relativistic regime where the speed of the photoelectron stays well below the velocity of the light $c$. We solve the time-dependent Schr\"odinger equation (TDSE) using a spectral approach with a basis of one-electron orbitals, calculated with ${L}^{2}$-integrable $B$-spline functions for the radial coordinate and bipolar spherical harmonics ${Y}_{lm}$ for the angular part. Retardation effects are included up to $O(1/c)$; they induce two extra terms leading to electric quadrupole and magnetic dipole transitions. These latter terms depend on the polarization and photon momentum directions, forcing the resolution of the TDSE in a three-dimensional space. Relativistic effects [of $O(1/{c}^{2})$] are fully neglected. Photoelectron energy and angular distributions are obtained for photon energies ranging from 200 eV to 3 keV. We study the lower energy region of the ATI spectrum and we analyze nondipole effects, through their $(l,m)$ components, in particular, for the lower two peaks. Although these effects are small, we show that, increasing the photon energy, the photoelectron angular distributions begin to differ significantly from the ones obtained in dipole approximation.

Odd-parity bipolar spherical harmonics

Bipolar spherical harmonics (BiPoSHs) provide a general formalism for quantifying departures in the cosmic microwave background (CMB) from statistical isotropy and from Gaussianity. However, prior work has focused only on BiPoSHs with even parity. Here we show that there is another set of BiPoSHs with odd parity, and we explore their cosmological applications. We describe systematic artifacts in a CMB map that could be sought by measurement of these odd-parity BiPoSH modes. These BiPoSH modes may also be produced cosmologically through lensing by gravitational waves (GWs), among other sources. We derive expressions for the BiPoSH modes induced by the weak lensing of both scalar and tensor perturbations. We then investigate the possibility of detecting parity-breaking physics, such as chiral GWs, by cross-correlating opposite-parity BiPoSH modes with multipole moments of the CMB polarization. We find that the expected signal-to-noise of such a detection is modest.

Spectral representation of the three-body Coulomb problem: Perspectives for highly doubly excited states of helium

We present a spectral method of configuration-interaction type for three-dimensional helium which combines the complex rotation method with an appropriate expansion of the atom wave function in a basis of products of Coulomb-Sturmian functions of the electron radial coordinates with independent dilation parameters for the two electrons and bipolar spherical harmonics of the angular coordinates. The matrix elements of the kinetic energy and of the electron-nucleus interaction term are calculated using Gauss-Laguerre integration techniques. A combination of Gauss-Laguerre integration techniques with the generalized Wigner-Eckart theorem and recurrence relations allows an efficient and stable calculation of the matrix elements of the electron-electron interaction. The freedom of the choice of the dilation parameters permits us to access highly excited states with rather small sizes of the basis. Highly doubly excited states up to the tenth ionization threshold of singlet and triplet S states of helium are presented.

Angular momentum coupling in NMR: a new view of rotation matrix products via Racah algebra

Choosing the rotation angle-rotation axis parameters instead of the conventional Euler angle parameters {α, β, γ} to parametrise rotations, and a spherical tensor operator basis, Happer was the first to propose a novel multipole operator expansion of the rotation operator [W. Happer, Ann. Phys. (N.Y.) 48, 579 (1968)]. As Happer pointed out, such an expansion in irreducible spherical tensor operators readily lends itself to the methods of Racah algebra, a feature we exploit to simplify the calculus of rotation matrix products. Working with a Clebsch–Gordan coefficient expansion [W. Happer, ibid. cit.; M.S. Marinov, Yad. Fiz. 5, 943 (1967)] of the corresponding rotation matrices , and closure relations for these (2J + 1)-dimensional irreducible representations of the rotation group, we state, and prove for the first time, simple recoupling coefficient expansions for rotation matrix products. With an eye towards applications in NMR, where multiple rotations in spin or coordinate space often require the efficient evaluation of such rotation matrix products, we focus on expansions defining the matrices representing the result of two or three consecutive rotations. We show that the coefficients of these expansions are defined by 6 − j symbols and bipolar spherical harmonics (two rotations), or by 9 − j symbols and tripolar spherical harmonics (three rotations). We show that these recoupling coefficient expansions, in conjunction with Racah algebra, are the cornerstone of an unusually simple and direct method of (1) proving the Euler–Rodrigues (quaternion) parameter composition rules for two and three consecutive rotations and (2) constructing rotational invariants expressed in Weyl-invariant form. These methods are distinguished by their independence on the usual algebraic approaches, either those of operator algebra, or ordinary algebra.