Higher Multipole Terms Research Articles

Testing general relativity with the Doppler magnification effect

ABSTRACT The apparent sizes and brightnesses of galaxies are correlated in a dipolar pattern around matter overdensities in redshift space, appearing larger on their near side and smaller on their far side. The opposite effect occurs for galaxies around an underdense region. These patterns of apparent magnification induce dipole and higher multipole terms in the cross-correlation of galaxy number density fluctuations with galaxy size/brightness (which is sensitive to the convergence field). This provides a means of directly measuring peculiar velocity statistics at low and intermediate redshift, with several advantages for performing cosmological tests of general relativity (GR). In particular, it does not depend on empirically calibrated scaling relations like the Tully–Fisher and Fundamental Plane methods. We show that the next generation of spectroscopic galaxy redshift surveys will be able to measure the Doppler magnification effect with sufficient signal-to-noise ratio to test GR on large scales. We illustrate this with forecasts for the constraints that can be achieved on parametrized deviations from GR for forthcoming low-redshift galaxy surveys with DESI and SKA2. Although the cross-correlation statistic considered has a lower signal-to-noise ratio than RSD, it will be a useful probe of GR since it is sensitive to different systematics.

Monopolar and quadrupolar gravitational radiation from magnetically deformed neutron stars in modified gravity

Some modified theories of gravity are known to predict monopolar, in addition to the usual quadrupolar and beyond, gravitational radiation in the form of `breathing' modes. For the same reason that octupole and higher-multipole terms often contribute negligibly to the overall wave strain, monopole terms tend to dominate. We investigate both monopolar and quadrupolar continuous gravitational radiation from neutron stars deformed through internal magnetic stresses. We adopt the Parameterised-Post-Newtonian formalism to write down equations describing the leading-order stellar properties in a theory-independent way, and derive some exact solutions for stars with mixed poloidal-toroidal magnetic fields. We then turn to the specific case of scalar-tensor theories to demonstrate how observational upper limits on the gravitational-wave luminosity of certain neutron stars may be used to place constraints on modified gravity parameters, most notably the Eddington parameter $\gamma$. For conservative, purely poloidal models with characteristic field strength given by the spindown minimum, upper limits for the Vela pulsar yield $1-\gamma \lesssim 4.2 \times 10^{-3}$. For models containing a strong toroidal field housing $\sim 99\%$ of the internal magnetic energy, we obtain the bound $1- \gamma \lesssim 8.0 \times 10^{-7}$. This latter bound is an order of magnitude tighter than those obtained from current Solar system experiments, though applies to the strong-field regime.

Treatment of U92+–U91+ collisions in spherical co-ordinates: going beyond the monopole approximation

Recently, a new approach has been developed to treat the time-dependent two-centre Dirac equation within our group. This method generates basis wavefunctions describing an electron in the field of two nuclei at fixed internuclear distances in spherical co-ordinates, and uses coupled-channel techniques in handling the time dependences. Using this method, we analyse the electronic properties accompanying a U92+–U91+ collision for a zero-impact parameter. Specifically, the ground state occupation probability and the ionization probability has been calculated. Based on our calculations, we show the importance of using higher multipole terms in the expansion of the two-centre potential.

Magnetic interconnection of Saturn's polar regions: comparison of modelling results with Hubble Space Telescope UV auroral images

Abstract. We consider the magnetic interconnection of Saturn's northern and southern polar regions controlled by the interplanetary magnetic field (IMF), studying in particular the more complex and interesting case of southward IMF, when the Kronian magnetospheric magnetic field structure is the most twisted. The simpler case of northward IMF is also discussed. Knowledge of the magnetospheric magnetic field structure is very significant, for example, for investigation of the electric fields and field-aligned currents in Saturn's environment, particularly those which cause the auroral emissions. Here we modify the paraboloid magnetospheric magnetic field model employed in previous related studies by including higher multipole terms in Saturn's internal magnetic field, required for more detailed considerations of inter-hemispheric conjugacy, together with inclusion of a spheroidal boundary at the ionospheric level. The model is employed to map Southern Hemisphere auroral regions observed by the Hubble Space Telescope (HST) in 2008 under known IMF conditions to both the equatorial plane and the northern ionosphere. It is shown that the brightest auroral features map typically to the equatorial region between the central ring current and the outer magnetosphere, and that auroral features should be largely symmetric between the two hemispheres, except for a small poleward displacement and latitudinal narrowing in the Northern Hemisphere compared with the Southern Hemisphere due to the quadrupole field asymmetry. The latter features are in agreement with the conjugate auroras observed under near-equinoctial conditions in early 2009, when IMF data are not available.

Structure of the asymptotic wavefunction for the N-body Coulomb problem

An analytic form for a wavefunction is presented which is an asymptotic solution to the N-body Schrodinger equation with the monopole part of the Coulomb interaction between all pairs of particles included. The structure of the wavefunction allows each two-body interaction to be treated separately so that higher multipole terms in the interaction can be systematically included. The results are presented for an unsymmetrised function, but symmetrisation of all or some of the particle coordinates follows immediately. The asymptotic region is defined by all radial coordinates becoming large but the ratios of radial coordinates for different particles are arbitrary.

The normal human magnetocardiogram. II. A multipole analysis.

This paper presents a multipole analysis of the QRS complex of the normal magnetocardiogram (MCG) of six normal subjects. The multipole strengths up to the octupole term were determined from the measured distribution of the z-component of the cardiac magnetic field (the component perpendicular to the frontal plane). This equivalent magnetic multipole generator was found to represent the cardiac magnetic field with a minimum error of less than 10%. The dipolar term of the expansion was found to represent the field distribution with reasonable accuracy early during ventricular activation, whereas late during activation, the higher multipole terms were clearly more significant.

Far-infrared absorption by electron-hole drops in germanium. Damping and size effects

Measurements of the far-infrared absorption of electron-hole drops in germanium at 1.2 K in the spectral range 1.5 to 50 meV are reported. The line shape agrees well with Mie's theory with a dielectric constant that includes interband hole transitions and a frequency-dependent damping. The absorption has a very sharp threshold at 2.8\ifmmode\pm\else\textpm\fi{}0.1 meV, the onset interband hole transitions. This sharpness is caused by an extremely long collision time for electron and holes near their incongruent Fermi surfaces. The electron-hole density is found to be (2.02\ifmmode\pm\else\textpm\fi{}0.05) \ifmmode\times\else\texttimes\fi{} ${10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. The line shape changes slightly with excitation level. At low excitation, the drops are smaller, have a lower average density, and the absorption is highly damped by surface collisions. A coated sphere model is used to include these effects in the Mie theory. At high excitation levels, the drops are larger and higher multipole terms in the Mie expansion must be included. It is found that drop sizes vary with excitation from 0.05 to 2 \ensuremath{\mu}m at 1.2 K.

Retarded dispersion forces between conducting spheres

The retarded dispersion forces between conducting spheres are calculated. It is shown that for perfect conductors, the contribution of magnetic dipole terms and higher-multipole terms are important, amounting to about 50% of the force asymptotically, and an even greater percentage at shorter distances. It is argued that ordinary conducting and superconducting spheres will behave as perfect conductors insofar as retarded dispersion forces are concerned, provided that the radius of the spheres is greater than about ${10}^{\ensuremath{-}4}$ cm for superconductors, and provided that the separation is less then ${10}^{2}a$ for ordinary conductors. Some comments on the measurability of these forces are presented.

Comments on the approximate calculation of lattice potential

The potential of a charge distribution due to a lattice of point charges may be evaluated by the classical multipole expansion method. The leading terms in the resultant expressions are just those used in some of our previous calculations [1–3]. In addition, for cases where the leading terms vanish because of the effect of orthogonality of the basis functions upon the Mulliken expansion (this being especially serious in the case of a one-centre charge distribution), we have derived the first nonvanishing term, involving 〈χμ|r|χν〉. In other cases it may be necessary to proceed to still higher multipole terms before a non-zero contribution is obtained. The entire procedure is formulated in such a way that it can be easily applied to LCAO-MO calculations for polyatomic ions in ionic lattices.

Infra-red spectra of crystals—IV. Matrix shifts and the contribution from classical electrostatic forces

Two principal defects of existing treatments of vibration frequency shifts in solution are discussed. Instead of representing the vibrating molecule by a point dipole, a more realistic description should be given by using a multipole expansion of the electrostatic potential energy. The results of multipole calculations are given for a simplified effective charge model, in a spherical cavity, appropriate to XY 4 molecules and the frequency shift data for Group IVB hydrides discussed. It is shown that the dipole-induced dipole term in the potential energy may amount to as little as 30 per cent of the total electrostatic contribution, the proportion being rather less for stretching than for bending modes. The point polarized atom model employed by Linevsky for LiF in inert gas matrices is compared with the cavity model and shown to predict much smaller shifts than the latter. Second order perturbation treatments are carried out to determine matrix elements for LiF and CO, and the very great contribution of the anharmonic terms to the frequency shift in LiF displayed. The calculated shift is even greater when the higher multipole terms appropriate to an effective charge model are included (∼190 cm −1 in Xe). The chief result of the increase in calculated negative shift due to classical electrostatic forces is to enhance the role played by repulsive forces. An alternative assignment is proposed for the two bands associated with CO in both Ar and Kr matrices.