Cosmology with Gravitational Waves: A Review

Standard sirens have been the central paradigm in gravitational-wave cosmology so far. From the gravitational wave signature of compact star binaries, it is possible to measure the luminosity distance of the source directly, and if additional information on the source redshift is provided, a measurement of the cosmological expansion can be performed. This review article discusses several methodologies that have been proposed to use gravitational waves for cosmological studies. Methods that use only gravitational-wave signals and methods that use gravitational waves in conjunction with additional observations such as electromagnetic counterparts and galaxy catalogs will be discussed. The review also discusses the most recent results on gravitational-wave cosmology, starting from the binary neutron star merger GW170817 and its electromagnetic counterpart and finishing with the population of binary black holes, observed with the third Gravitational-wave Transient Catalog GWTC–3.     

Numerical Computation of Galois Groups

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators, while the other studies its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.     

Parity-induced Selmer growth for symplectic, ordinary families

Let p be an odd prime, and let K/K₀ be a quadratic extension of number fields. Denote by K_± the maximal Zp-power extensions of K that are Galois over K₀, with K₊ abelian over K₀ and K₋ dihedral over K₀. In this paper we show that for a Galois representation over K₀ satisfying certain hypotheses, if it has odd Selmer rank over K then for one of K_± its Selmer rank over L is bounded below by [L:K] for L ranging over the finite subextensions of K in K_±. Our method of proof generalizes a method of Mazur and Rubin, building upon results of Neková?, and applies to abelian varieties of arbitrary dimension, (self-dual twists of) modular forms of even weight, and (twisted) Hida families.     

In situ studies of the active sites for the water gas shift reaction over Cu-CeO2 catalysts: complex interaction between metallic copper and oxygen vacancies of ceria

New information about the active sites for the water gas shift (WGS) reaction over Cu-CeO2 systems was obtained using in-situ, time-resolved X-ray diffraction (TR-XRD), X-ray absorption spectroscopy (TR-XAS, Cu K and Ce L3 edges), and infrared spectroscopy (DRIFTS). Cu-CeO2 nanoparticles prepared by a novel reversed microemulsion method (doped Ce1-xCuxO2 sample) and an impregnation method (impregnated CuOx/CeO2 sample) were studied. The results from all of the samples indicate that both metallic copper and oxygen vacancies in ceria were involved in the generation of active sites for the WGS reaction. Evidence was found for a synergistic Cu-Ovacancy interaction. This interaction enhances the chemical activity of Cu, and the presence of Cu facilitates the formation of O vacancies in ceria under reaction conditions. Water dissociation occurred on the Ovacancy sites or the Cu-Ovacancy interface. No significant amounts of formate were formed on the catalysts during the WGS reaction. The presence of strongly bound carbonates is an important factor for the deactivation of the catalysts at high temperatures. This work identifies for the first time the active sites for the WGS reaction on Cu-CeO2 catalysts and illustrates the importance of in situ structural studies for heterogeneous catalytic reactions.     

L 1 factorizations and invariant subspaces for weighted composition operators

For a contraction operator T with spectral radius less than one on a Banach space , it is shown that the factorization of certain L¹ functions by vectors x in and x*. in , in the sense that for n ≧ 0, implies the existence of invariant subspaces for T. Explicit formulae for such factorizations are given in the case of weighted composition operators on reproducing kernel Hilbert spaces. An interpolation result of McPhail is applied to show how this can be used to construct invariant subspaces of hyperbolic weighted composition operators on H². Received: 1 November 2005     

Anomalous Richtmyer-Meshkov fingering in dissipative particle systems

We study the fingering instability induced by a shock that propagates across a perturbed interface that separates two types of discrete particles. If collisions between particles conserve energy, then the relative sizes and growth rates of the fingers are similar to those in the analogous shock-induced fingering instability in fluids. However, we show that energy loss during particle collisions, even when very small, causes the qualitative features of the finger growth to be completely opposite to the fluid case. The fingers formed by light particles grow faster and become longer and narrower than the fingers formed by heavy particles. In addition, the finger composed of light particles collapses into an extremely compact, tortuous filament, and diffusive mixing between particle types at the particle scale is heavily suppressed.     

Improved Decay for Solutions to the Linear Wave Equation on a Schwarzschild Black Hole

We prove that sufficiently regular solutions to the wave equation ${\square_g\phi=0}We prove that sufficiently regular solutions to the wave equation \square_gf = 0{\square_g\phi=0} on the exterior of the Schwarzschild black hole obey the estimates |f| ￡ C_d v₊^-\frac32+d{|\phi|\leq C_\delta v_+^{-\frac{3}{2}+\delta}} and |?_tf| ￡ C_d v₊^-2+d{|\partial_t\phi|\leq C_{\delta} v_+^{-2+\delta}} on a compact region of r, including inside the black hole region. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite r and along the event horizon.     

A generalized approach to the calculation of ro-vibrational spectra of triatomic molecules

A generalization of the well known atom-diatom scattering hamiltonian to a coordinate system of two lengths and an angle is derived, another special case of which is a previously known bond angle-bond length hamiltonian. Different axis embeddings are also considered. The formalism is applied to the ro-vibrational levels of D₂H⁺, CH⁺ ₂ and HDHe (A ¹ A′) and the advantage of a judicious choice of coordinates demonstrated. The vibrational band origins for HDHe*, the first predictions for this system for which previous calculations had failed, are obtained using a new geometrically defined coordinate system. It is suggested that these coordinates might be used to represent isotopically substituted van der Waals complexes.