2,510 research outputs found
Quasi-periodic solutions of completely resonant forced wave equations
We prove existence of quasi-periodic solutions with two frequencies of
completely resonant, periodically forced nonlinear wave equations with periodic
spatial boundary conditions. We consider both the cases the forcing frequency
is: (Case A) a rational number and (Case B) an irrational number.Comment: 25 pages, 1 figur
Constraining properties of the black hole population using LISA
LISA should detect gravitational waves from tens to hundreds of systems
containing black holes with mass in the range from 10 thousand to 10 million
solar masses. Black holes in this mass range are not well constrained by
current electromagnetic observations, so LISA could significantly enhance our
understanding of the astrophysics of such systems. In this paper, we describe a
framework for combining LISA observations to make statements about massive
black hole populations. We summarise the constraints that LISA observations of
extreme-mass-ratio inspirals might be able to place on the mass function of
black holes in the LISA range. We also describe how LISA observations can be
used to choose between different models for the hierarchical growth of
structure in the early Universe. We consider four models that differ in their
prescription for the initial mass distribution of black hole seeds, and in the
efficiency of accretion onto the black holes. We show that with as little as 3
months of LISA data we can clearly distinguish between these models, even under
relatively pessimistic assumptions about the performance of the detector and
our knowledge of the gravitational waveforms.Comment: 12 pages, 3 figures, submitted to Class. Quantum Grav. for
proceedings of 8th LISA Symposium; v2 minor changes for consistency with
accepted versio
LISA observations of massive black hole mergers: event rates and issues in waveform modelling
The observability of gravitational waves from supermassive and
intermediate-mass black holes by the forecoming Laser Interferometer Space
Antenna (LISA), and the physics we can learn from the observations, will depend
on two basic factors: the event rates for massive black hole mergers occurring
in the LISA best sensitivity window, and our theoretical knowledge of the
gravitational waveforms. We first provide a concise review of the literature on
LISA event rates for massive black hole mergers, as predicted by different
formation scenarios. Then we discuss what (in our view) are the most urgent
issues to address in terms of waveform modelling. For massive black hole binary
inspiral these include spin precession, eccentricity, the effect of high-order
Post-Newtonian terms in the amplitude and phase, and an accurate prediction of
the transition from inspiral to plunge. For black hole ringdown, numerical
relativity will ultimately be required to determine the relative quasinormal
mode excitation, and to reduce the dimensionality of the template space in
matched filtering.Comment: 14 pages, 2 figures. Added section with conclusions and outlook.
Matches version to appear in the proceedings of 10th Annual Gravitational
Wave Data Analysis Workshop (GWDAW 10), Brownsville, Texas, 14-17 Dec 200
Periodic solutions for a class of nonlinear partial differential equations in higher dimension
We prove the existence of periodic solutions in a class of nonlinear partial
differential equations, including the nonlinear Schroedinger equation, the
nonlinear wave equation, and the nonlinear beam equation, in higher dimension.
Our result covers cases where the bifurcation equation is infinite-dimensional,
such as the nonlinear Schroedinger equation with zero mass, for which solutions
which at leading order are wave packets are shown to exist.Comment: 34 page
Convergence in measure under Finite Additivity
We investigate the possibility of replacing the topology of convergence in
probability with convergence in . A characterization of continuous linear
functionals on the space of measurable functions is also obtained
Time quasi-periodic gravity water waves in finite depth
We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions\u2014namely periodic and even in the space variable x\u2014of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the fully nonlinear nature of the gravity water waves equations\u2014the highest order x-derivative appears in the nonlinear term but not in the linearization at the origin\u2014and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators, obtained at each approximate quasi-periodic solution along a Nash\u2013Moser iterative scheme, to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory
Perturbative calculation of quasi-normal modes of Schwarzschild black holes
We discuss a systematic method of analytically calculating the asymptotic
form of quasi-normal frequencies of a four-dimensional Schwarzschild black hole
by expanding around the zeroth-order approximation to the wave equation
proposed by Motl and Neitzke. We obtain an explicit expression for the
first-order correction and arbitrary spin. Our results are in agreement with
the results from WKB and numerical analyses in the case of gravitational waves.Comment: 11 pages; references added and a sign error corrected; to appear in
CQ
Quasinormal ringing of Kerr black holes: The excitation factors
Distorted black holes radiate gravitational waves. In the so-called ringdown
phase radiation is emitted in a discrete set of complex quasinormal
frequencies, whose values depend only on the black hole's mass and angular
momentum. Ringdown radiation could be detectable with large signal-to-noise
ratio by the Laser Interferometer Space Antenna LISA. If more than one mode is
detected, tests of the black hole nature of the source become possible. The
detectability of different modes depends on their relative excitation, which in
turn depends on the cause of the perturbation (i.e. on the initial data). A
``universal'', initial data-independent measure of the relative mode excitation
is encoded in the poles of the Green's function that propagates small
perturbations of the geometry (``excitation factors''). We compute for the
first time the excitation factors for general-spin perturbations of Kerr black
holes. We find that for corotating modes with the excitation factors tend
to zero in the extremal limit, and that the contribution of the overtones
should be more significant when the black hole is fast rotating. We also
present the first analytical calculation of the large-damping asymptotics of
the excitation factors for static black holes, including the Schwarzschild and
Reissner-Nordstrom metrics. This is an important step to determine the
convergence properties of the quasinormal mode expansion.Comment: 33 pages, 9 figures, 7 tables, RevTeX4. v2: Two new figures and minor
changes in the presentation. Matches version in press in Phys. Rev.
Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes
We establish a Bohr-Sommerfeld type condition for quasi-normal modes of a
slowly rotating Kerr-de Sitter black hole, providing their full asymptotic
description in any strip of fixed width. In particular, we observe a
Zeeman-like splitting of the high multiplicity modes at a=0 (Schwarzschild-de
Sitter), once spherical symmetry is broken. The numerical results presented in
Appendix B show that the asymptotics are in fact accurate at very low energies
and agree with the numerical results established by other methods in the
physics literature. We also prove that solutions of the wave equation can be
asymptotically expanded in terms of quasi-normal modes; this confirms the
validity of the interpretation of their real parts as frequencies of
oscillations, and imaginary parts as decay rates of gravitational waves.Comment: 66 pages, 6 figures; journal version (to appear in Annales Henri
Poincar\'e
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