Hadamard Regularization

Motivated by the problem of the dynamics of point-particles in high post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a certain class of functions which are smooth except at some isolated points around which they admit a power-like singular expansion. We review the concepts of (i) Hadamard ``partie finie'' of such functions at the location of singular points, (ii) the partie finie of their divergent integral. We present and investigate different expressions, useful in applications, for the latter partie finie. To each singular function, we associate a partie-finie (Pf) pseudo-function. The multiplication of pseudo-functions is defined by the ordinary (pointwise) product. We construct a delta-pseudo-function on the class of singular functions, which reduces to the usual notion of Dirac distribution when applied on smooth functions with compact support. We introduce and analyse a new derivative operator acting on pseudo-functions, and generalizing, in this context, the Schwartz distributional derivative. This operator is uniquely defined up to an arbitrary numerical constant. Time derivatives and partial derivatives with respect to the singular points are also investigated. In the course of the paper, all the formulas needed in the application to the physical problem are derived.Comment: 50 pages, to appear in Journal of Mathematical Physic

Gravitational-Wave Recoil from the Ringdown Phase of Coalescing Black Hole Binaries

The gravitational recoil or "kick" of a black hole formed from the merger of two orbiting black holes, and caused by the anisotropic emission of gravitational radiation, is an astrophysically important phenomenon. We combine (i) an earlier calculation, using post-Newtonian theory, of the kick velocity accumulated up to the merger of two non-spinning black holes, (ii) a "close-limit approximation" calculation of the radiation emitted during the ringdown phase, and based on a solution of the Regge-Wheeler and Zerilli equations using initial data accurate to second post-Newtonian order. We prove that ringdown radiation produces a significant "anti-kick". Adding the contributions due to inspiral, merger and ringdown phases, our results for the net kick velocity agree with those from numerical relativity to 10-15 percent over a wide range of mass ratios, with a maximum velocity of 180 km/s at a mass ratio of 0.38.Comment: 9 pages, 5 figures; to appear in Class. Quant. Gra

Determination of Dark Energy by the Einstein Telescope: Comparing with CMB, BAO and SNIa Observations

A design study is currently in progress for a third generation gravitational-wave (GW) detector called Einstein Telescope (ET). An important kind of source for ET will be the inspiral and merger of binary neutron stars (BNS) up to $z \sim 2$. If BNS mergers are the progenitors of short-hard $\gamma$-ray bursts, then some fraction of them will be seen both electromagnetically and through GW, so that the luminosity distance and the redshift of the source can be determined separately. An important property of these `standard sirens' is that they are \emph{self-calibrating}: the luminosity distance can be inferred directly from the GW signal, with no need for a cosmic distance ladder. Thus, standard sirens will provide a powerful independent check of the $\Lambda$CDM model. In previous work, estimates were made of how well ET would be able to measure a subset of the cosmological parameters (such as the dark energy parameter $w_0$) it will have access to, assuming that the others had been determined to great accuracy by alternative means. Here we perform a more careful analysis by explicitly using the potential Planck CMB data as prior information for these other parameters. We find that ET will be able to constrain $w_0$ and $w_a$ with accuracies $\Delta w_0 = 0.099$ and $\Delta w_a = 0.302$, respectively. These results are compared with projected accuracies for the JDEM Baryon Acoustic Oscillations project and the SNAP Type Ia supernovae observations.Comment: 28 pages, 5 figures, 5 tables; Published Versio

Propagation of gravitational waves from slow motion sources in a Coulomb type potential

We consider the propagation of gravitational waves generated by slow motion sources in Coulomb type potential due to the mass of the source. Then, the formula for gravitational waveform including tail is obtained in a straightforward manner by using the spherical Coulomb function. We discuss its relation with the formula in the previous work.Comment: 13 pages, no figures, to be published in Phys. Rev.

Factorization in Formal Languages

We consider several novel aspects of unique factorization in formal languages. We reprove the familiar fact that the set uf(L) of words having unique factorization into elements of L is regular if L is regular, and from this deduce an quadratic upper and lower bound on the length of the shortest word not in uf(L). We observe that uf(L) need not be context-free if L is context-free. Next, we consider variations on unique factorization. We define a notion of "semi-unique" factorization, where every factorization has the same number of terms, and show that, if L is regular or even finite, the set of words having such a factorization need not be context-free. Finally, we consider additional variations, such as unique factorization "up to permutation" and "up to subset"

Classicalization of Gravitons and Goldstones

We establish a close parallel between classicalization of gravitons and derivatively-coupled Nambu-Goldstone-type scalars. We show, that black hole formation in high energy scattering process represents classicalization with the classicalization radius given by Schwarzschild radius of center of mass energy, and with the precursor of black hole entropy being given by number of soft quanta composing this classical configuration. Such an entropy-equivalent is defined for scalar classicalons also and is responsible for exponential suppression of their decay into small number of final particles. This parallel works in both ways. For optimists that are willing to hypothesize that gravity may indeed self-unitarize at high energies via black hole formation, it illustrates that the Goldstones may not be much different in this respect, and they classicalize essentially by similar dynamics as gravitons. In the other direction, it may serve as an useful de-mystifier of via-black-hole-unitarization process and of the role of entropy in it, as it illustrates, that much more prosaic scalar theories essentially do the same. Finally, it illustrates that in both cases classicalization is the defining property for unitarization, and that it sets-in before one can talk about accompanying properties, such as entropy and thermality of static classicalons (black holes). These properties are by-products of classicalization, and their equivalents can be defined for non-gravitational cases of classicalization.Comment: 23 page

The self-consistent gravitational self-force

I review the problem of motion for small bodies in General Relativity, with an emphasis on developing a self-consistent treatment of the gravitational self-force. An analysis of the various derivations extant in the literature leads me to formulate an asymptotic expansion in which the metric is expanded while a representative worldline is held fixed; I discuss the utility of this expansion for both exact point particles and asymptotically small bodies, contrasting it with a regular expansion in which both the metric and the worldline are expanded. Based on these preliminary analyses, I present a general method of deriving self-consistent equations of motion for arbitrarily structured (sufficiently compact) small bodies. My method utilizes two expansions: an inner expansion that keeps the size of the body fixed, and an outer expansion that lets the body shrink while holding its worldline fixed. By imposing the Lorenz gauge, I express the global solution to the Einstein equation in the outer expansion in terms of an integral over a worldtube of small radius surrounding the body. Appropriate boundary data on the tube are determined from a local-in-space expansion in a buffer region where both the inner and outer expansions are valid. This buffer-region expansion also results in an expression for the self-force in terms of irreducible pieces of the metric perturbation on the worldline. Based on the global solution, these pieces of the perturbation can be written in terms of a tail integral over the body's past history. This approach can be applied at any order to obtain a self-consistent approximation that is valid on long timescales, both near and far from the small body. I conclude by discussing possible extensions of my method and comparing it to alternative approaches.Comment: 44 pages, 4 figure

An approximate binary-black-hole metric

An approximate solution to Einstein's equations representing two widely-separated non-rotating black holes in a circular orbit is constructed by matching a post-Newtonian metric to two perturbed Schwarzschild metrics. The spacetime metric is presented in a single coordinate system valid up to the apparent horizons of the black holes. This metric could be useful in numerical simulations of binary black holes. Initial data extracted from this metric have the advantages of being linked to the early inspiral phase of the binary system, and of not containing spurious gravitational waves.Comment: 20 pages, 1 figure; some changes in Sec. IV B,C and Sec.

Gravitational waveforms from inspiralling compact binaries to second-post-Newtonian order

The two independent ``plus" and ``cross" polarization waveforms associated with the gravitational waves emitted by inspiralling, non-spinning, compact binaries are presented, ready for use in the data analysis of signals received by future laser interferometer gravitational-wave detectors such as LIGO and VIRGO. The computation is based on a recently derived expression of the gravitational field at the second-post-Newtonian approximation of general relativity beyond the dominant (Newtonian) quadrupolar field. The use of these theoretical waveforms to make measurements of astrophysical parameters and to test the nature of relativistic gravity is discussed.Comment: 17 pages; To appear in Classical and Quantum Gravit