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Register a new userA two-parameter family of double-power-law biorthonormal potential-density expansions
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Biorthonormal basis function expansions are widely used in galactic dynamics, both to study problems in galactic stability and to provide numerical algorithms to evolve collisionless stellar systems. They also provide a compact and efficient description of the structure of numerical dark matter haloes in cosmological simulations. We present a two-parameter family of biorthonormal double-power-law potential-density expansions. Both the potential and density are given in closed analytic form and may be rapidly computed via recurrence relations. We show that this family encompasses all the known analytic biorthonormal expansions: the Zhao expansions (themselves generalizations of ones found earlier by Hernquist & Ostriker and by Clutton-Brock) and the recently discovered Lilley, Sanders, Evans & Erkal expansion. Our new two-parameter family includes expansions based around many familiar spherical density profiles as zeroth-order models, including the $gamma$ models and the Jaffe model. It also contains a basis expansion that reproduces the famous Navarro-Frenk-White (NFW) profile at zeroth order. The new basis expansions have been found via a systematic methodology which has wide applications in finding further examples. In the process, we also uncovered a novel integral transform solution to Poissons equation.
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Efficient expansions of the gravitational field of (dark) haloes have two main uses in the modelling of galaxies: first, they provide a compact representation of numerically-constructed (or real) cosmological haloes, incorporating the effects of triaxiality, lopsidedness or other distortion. Secondly, they provide the basis functions for self-consistent field expansion algorithms used in the evolution of $N$-body systems. We present a new family of biorthogonal potential-density pairs constructed using the Hankel transform of the Laguerre polynomials. The lowest-order density basis functions are double-power-law profiles cusped like $rho sim r^{-2 + 1/alpha}$ at small radii with asymptotic density fall-off like $rho sim r^{-3 -1/(2alpha)}$. Here, $alpha$ is a parameter satisfying $alpha ge 1/2$. The family therefore spans the range of inner density cusps found in numerical simulations, but has much shallower -- and hence more realistic -- outer slopes than the corresponding members of the only previously-known family deduced by Zhao (1996) and exemplified by Hernquist & Ostriker (1992). When $alpha =1$, the lowest-order density profile has an inner density cusp of $rho sim r^{-1}$ and an outer density slope of $rho sim r^{-3.5}$, similar to the famous Navarro, Frenk & White (1997) model. For this reason, we demonstrate that our new expansion provides a more accurate representation of flattened NFW haloes than the competing Hernquist-Ostriker expansion. We utilize our new expansion by analysing a suite of numerically-constructed haloes and providing the distributions of the expansion coefficients.
Galaxy kinematics and gravitational lensing are two complementary ways to constrain the distribution of dark matter on galaxy scales. The typical dark matter density profiles adopted in dynamical studies cannot easily be adopted in lensing studies. Ideally, a mass model should be used that has the global characteristics of realistic dark matter distributions, and that allows for an analytical calculation of the magnifications and deflection angles. A simple model with these properties, the broken-power-law (BPL) model, has very recently been introduced. We examine the dynamical structure of the family of BPL models. We derive simple closed expressions for basic dynamical properties, and study the distribution function under the assumption of velocity isotropy. We find that none of the BPL models with realistic parameters has an isotropic distribution function that is positive over the entire phase space, implying that the BPL models cannot be supported by an isotropic velocity distribution, or models with a more radially anisotropic orbital structure. This result limits the attractiveness of the BPL family as a tool for lensing studies to some degree. More generally, we find that not all members of the general family of double power-law or Zhao models, often used to model dark matter haloes, can be supported by an isotropic or radially anisotropic distribution function. In other words, the distribution function may become negative even for spherically symmetric models with a well-behaved density profile.
Inspired by an interesting counterexample to the cosmic no-hair conjecture found in a supergravity-motivated model recently, we propose a multi-field extension, in which two scalar fields are allowed to non-minimally couple to two vector fields, respectively. This model is shown to admit an exact Bianchi type I power-law solution. Furthermore, stability analysis based on the dynamical system method is performed to show that this anisotropic solution is indeed stable and attractive if both scalar fields are canonical. Nevertheless, if one of the two scalar fields is phantom then the corresponding anisotropic power-law inflation turns unstable as expected.
We report a simple method to generate potential/surface density pairs in flat axially symmetric finite size disks. Potential/surface density pairs consist of a ``homogeneous pair (a closed form expression) corresponding to a uniform disk, and a ``residual pair. This residual component is converted into an infinite series of integrals over the radial extent of the disk. For a certain class of surface density distributions (like power laws of the radius), this series is fully analytical. The extraction of the homogeneous pair is equivalent to a convergence acceleration technique, in a matematical sense. In the case of power law distributions, the convergence rate of the residual series is shown to be cubic inside the source. As a consequence, very accurate potential values are obtained by low order truncation of the series. At zero order, relative errors on potential values do not exceed a few percent typically, and scale with the order N of truncation as 1/N**3. This method is superior to the classical multipole expansion whose very slow convergence is often critical for most practical applications.
Star-forming galaxies display a close relation among stellar mass, metallicity and star-formation rate (or molecular-gas mass). This is known as the fundamental metallicity relation (FMR) (or molecular-gas FMR), and it has a profound implication on models of galaxy evolution. However, there still remains a significant residual scatter around the FMR. We show here that a fourth parameter, the surface density of stellar mass, reduces the dispersion around the molecular-gas FMR. In a principal component analysis of 29 physical parameters of 41,338 star-forming galaxies, the surface density of stellar mass is found to be the fourth most important parameter. The new four-dimensional (4D) fundamental relation forms a tighter hypersurface that reduces the metallicity dispersion to 50% of that of the molecular-gas FMR. We suggest that future analyses and models of galaxy evolution should consider the FMR in a 4D space that includes surface density. The dilution time scale of gas inflow and the star-formation efficiency could explain the observational dependence on surface density of stellar mass. AKARI is expected to play an important role in shedding light on the infrared properties of the new 4D FMR.
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