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Register a new userOdd-Parity Bipolar Spherical Harmonics
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Authors
Laura G. Book
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Bipolar spherical harmonics (BiPoSHs) provide a general formalism for quantifying departures in the cosmic microwave background (CMB) from statistical isotropy (SI) 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.
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Cosmological observations are promising ways to improve our understanding of neutrino mass properties. The upper bound on the sum of masses is given by the cosmic microwave background and large scale structure. These measurements are all parity-even, and potentially contaminated by unmodeled baryonic effects. In this paper we propose a novel parity-odd gravitational effect of neutrinos: A unique contribution to the directions of the angular momentum field of galaxies and halos. This observable is free of contamination in linear perturbation theory, and thus likely more cleanly separated from other nongravitational effects. A deep 21-cm survey to redshift 1 can potentially yield a $5sigma$ significance on neutrino mass detection for a fiducial sum of neutrino masses of 0.05 eV.
Spherical Harmonics, $Y_ell^m(theta,phi)$, are derived and presented (in a Table) for half-odd-integer values of $ell$ and $m$. These functions are eigenfunctions of $L^2$ and $L_z$ written as differential operators in the spherical-polar angles, $theta$ and $phi$. The Fermion Spherical Harmonics are a new, scalar and angular-coordinate-dependent representation of fermion spin angular momentum. They have $4pi$ symmetry in the angle $phi$, and hence are not single-valued functions on the Euclidean unit sphere; they are double-valued functions on the sphere, or alternatively are interpreted as having a double-sphere as their domain.
We compare the statistics of parity even and odd multipoles of the cosmic microwave background (CMB) sky from PLANCK full mission temperature measurements. An excess power in odd multipoles compared to even multipoles has previously been found on large angular scales. Motivated by this apparent parity asymmetry, we evaluate directional statistics associated with even compared to odd multipoles, along with their significances. Primary tools are the emph{Power Tensor} and emph{Alignment Tensor} statistics. We limit our analysis to the first sixty multipoles i.e., $l=[2,61]$. We find no evidence for statistically unusual alignments of even parity multipoles. More than one independent statistic finds evidence for alignments of anisotropy axes of odd multipoles, with a significance equivalent to $sim 2 sigma$ or more. The robustness of alignment axes is tested by making galactic cuts and varying the multipole range. Very interestingly, the region spanned by the (a)symmetry axes is found to broadly contain other parity (a)symmetry axes previously observed in the literature.
The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom. It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students. In this work, we show how one can determine spherical harmonics in a more natural way based on operators and a powerful identity called the exponential disentangling operator identity (known in quantum optics, but little used elsewhere). This new strategy follows naturally after one has introduced Dirac notation, computed the angular momentum algebra, and determined the action of the angular momentum raising and lowering operators on the simultaneous angular momentum eigenstates (under $hat L^2$ and $hat L_z$).
Photometry of short-period planetary systems allows astronomers to monitor exoplanets, their host stars, and their mutual interactions. In addition to the transits of a planet in front of its star and the eclipses of the planet by its star, researchers have reported flux variations at the orbital frequency and its harmonics: planetary reflection and/or emission and Doppler beaming of starlight produce one peak per orbit, while ellipsoidal variations of a tidally distorted star and/or planet produce two maxima per orbit. Researchers have also reported significant photometric variability at three times the orbital frequency, sometimes much greater than the predictions of tidal theory. The reflected phase variations of a homogeneous planet contains power at even orbital harmonics-important for studies of ellipsoidal variations-but cannot contain odd orbital harmonics. We show that odd harmonics can, however, be produced by an edge-on planet with a time-variable map, or an inclined planet with a North-South (N-S) asymmetric map. Either of these scenarios entail weather: short-period planets are expected to have zero obliquity and hence N-S symmetric stellar forcing. North-South asymmetry in a giant planet would therefore suggest stochastic localized features, such as weather. However, we find that previous claims of large-amplitude odd modes in Kepler photometry are artifacts of removing planetary transits rather than modeling them. The only reliable claims of odd harmonics remain HAT-P-7b and Kepler-13Ab, for which the third mode amplitude is 6-8% of the planetary flux. Although time-variable albedo maps could in principle explain these odd harmonics, upper-limits on the infrared variability of hot Jupiters make this hypothesis unlikely. We urge theorists to study the effects of close-in planets on stellar atmospheres, as this remains the only plausible hypothesis.
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