arXiv subjects

Ryuichi Fujita

Publications and source records attributed to Ryuichi Fujita.

26 records · Page 2Linked to original sources

arXiv2013

Impact of the second order self-forces on the dephasing of the gravitational waves from quasi-circular extreme mass-ratio inspirals

The accurate calculation of long-term phase evolution of gravitational wave (GW) forms from extreme (intermediate) mass ratio inspirals (E(I)MRIs) is an inevitable step to extract information from this system. Achieving this goal, it is believed that we need to understand the gravitational self-forces. However, it is not quntatively demonstrated that the second order self-forces are necessary for this purpose. In this paper we revisit the problem to estimate the order of magnitude of the dephasing caused by the second order self-forces on a small body in a quasi-circular orbit around a Kerr black hole, based on the knowledge of the post-Newtonian (PN) approximation and invoking the energy balance argument. In particular, we focus on the averaged dissipative part of the self-force, since it gives the leading order contribution among the various components of them. To avoid the possibility that the energy flux of GWs becomes negative, we propose a new simple resummation called exponential resummation, which assures the positivity of the energy flux. In order to estimate the magnitude of the yet unknown second order self-forces, here we point out the scaling property in the absolute value of the PN coefficients of the energy flux. Using these new tools, we evaluate the expected magnitude of dephasing. Our analysis indicates that the dephasing due to the second order self-forces for quasi-circular E(I)MRIs may be well captured by the 3PN energy flux, once we obtain all the spin dependent terms, except for the case with an extremely large spin of the central Kerr black.

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arXiv2012

Gravitational Waves from a Particle in Circular Orbits around a Schwarzschild Black Hole to the 22nd Post-Newtonian Order

We extend our previous results of the 14th post-Newtonian (PN) order expansion of gravitational waves for a test particle in circular orbits around a Schwarzschild black hole to the 22PN order, i.e. $v^{44}$ beyond the leading Newtonian approximation where $v$ is the orbital velocity of a test particle. Comparing our 22PN formula for the energy flux with high precision numerical results, we find that the relative error of the 22PN flux at the innermost stable circular orbit is about $10^{-5}$. We also estimate the phase difference between the 22PN waveforms and numerical waveforms after a two-year inspiral. We find that the dephase is about $10^{-9}$ for $μ/M=10^{-4}$ and $10^{-2}$ for $μ/M=10^{-5}$ where $μ$ is the mass of the compact object and $M$ the mass of the central supermassive black hole. Finally, we construct a hybrid formula of the energy flux by supplementing the 4PN formula of the energy flux for circular and equatorial orbits around a Kerr black hole with all the present 22PN terms for the case of a Schwarzschild black hole. Comparing the hybrid formula with the the full numerical results, we examine the performance of the hybrid formula for the case of Kerr black hole.

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arXiv2012

Analytical solutions of bound timelike geodesic orbits in Kerr spacetime

We derive the analytical solutions of the bound timelike geodesic orbits in Kerr spacetime. The analytical solutions are expressed in terms of the elliptic integrals using Mino time $λ$ as the independent variable. Mino time decouples the radial and polar motion of a particle and hence leads to forms more useful to estimate three fundamental frequencies, radial, polar and azimuthal motion, for the bound timelike geodesics in Kerr spacetime. This paper gives the first derivation of the analytical expressions of the fundamental frequencies. This paper also gives the first derivation of the analytical expressions of all coordinates for the bound timelike geodesics using Mino time. These analytical expressions should be useful not only to investigate physical properties of Kerr geodesics but more importantly to applications related to the estimation of gravitational waves from the extreme mass ratio inspirals.

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arXiv2012

Gravitational radiation for extreme mass ratio inspirals to the 14th post-Newtonian order

We derive gravitational waveforms needed to compute the 14th post-Newtonian (14PN) order energy flux for a particle in circular orbit around a Schwarzschild black hole, i.e. $v^{28}$ beyond the leading Newtonian approximation where $v$ is the orbital velocity of a test particle. We investigate the convergence of the energy flux in the PN expansion and suggest a fitting formula which can be used to extract unknown higher order PN coefficients from accurate numerical data for more general orbits around a Kerr black hole. The phase difference between the 14PN waveforms and numerical waveforms after two years inspiral is shown to be about $10^{-7}$ for $μ/M=10^{-4}$ and $10^{-3}$ for $μ/M=10^{-5}$ where $μ$ is the mass of a compact object and $M$ the mass of the central supermassive black hole. In first order black hole perturbation theory, for extreme mass ratio inspirals which are one of the main targets of Laser Interferometer Space Antenna, the 14PN expressions will lead to the data analysis accuracies comparable to the ones resulting from high precision numerical waveforms.

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arXiv2010

Spherical harmonic modes of 5.5 post-Newtonian gravitational wave polarizations and associated factorized resummed waveforms for a particle in circular orbit around a Schwarzschild black hol

Recent breakthroughs in numerical relativity enable one to examine the validity of the post-Newtonian expansion in the late stages of inspiral. For the comparison between post-Newtonian (PN) expansion and numerical simulations, the waveforms in terms of the spin-weighted spherical harmonics are more useful than the plus and cross polarizations, which are used for data analysis of gravitational waves. Factorized resummed waveforms achieve better agreement with numerical results than the conventional Taylor expanded post-Newtonian waveforms. In this paper, we revisit the post-Newtonian expansion of gravitational waves for a test-particle of mass $\m$ in circular orbit of radius $r_0$ around a Schwarzschild black hole of mass $M$ and derive the spherical harmonic components associated with the gravitational wave polarizations up to order $v^{11}$ beyond Newtonian. Using the more accurate $h_{\ell m}$'s computed in this work, we provide the more complete set of associated $ρ_{\ell m}$'s and $δ_{\ell m}$'s that form important bricks in the factorized resummation of waveforms with potential applications for the construction of further improved waveforms for prototypical compact binary sources in the future. We also provide ready-to-use expressions of the 5.5PN gravitational waves polarizations $h_+$ and $h_\times$ in the test-particle limit for gravitational wave data analysis applications. Additionally, we provide closed analytical expressions for 2.5PN $h_{\ell m}$, 2PN $ρ_{\ell m}$ and 3PN $δ_{\ell m}$, for general multipolar orders $\ell$ and $m$ in the test-particle limit. Finally, we also examine the implications of the present analysis for compact binary sources in Laser Interferometer Space Antenna.

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arXiv2009

An Efficient Numerical Method for Computing Gravitational Waves Induced by a Particle Moving on Eccentric Inclined Orbits around a Kerr Black Hole

We develop a numerical code to compute gravitational waves induced by a particle moving on eccentric inclined orbits around a Kerr black hole. For such systems, the black hole perturbation method is applicable. The gravitational waves can be evaluated by solving the Teukolsky equation with a point like source term, which is computed from the stress-energy tensor of a test particle moving on generic bound geodesic orbits. In our previous papers, we computed the homogeneous solutions of the Teukolsky equation using a formalism developed by Mano, Suzuki and Takasugi and showed that we could compute gravitational waves efficiently and very accurately in the case of circular orbits on the equatorial plane. Here, we apply this method to eccentric inclined orbits. The geodesics around a Kerr black hole have three constants of motion: energy, angular momentum and the Carter constant. We compute the rates of change of the Carter constant as well as those of energy and angular momentum. This is the first time that the rate of change of the Carter constant has been evaluated accurately. We also treat the case of highly eccentric orbits with $e=0.9$. To confirm the accuracy of our codes, several tests are performed. We find that the accuracy is only limited by the truncation of $\ell$-, $k$- and $n$-modes, where $\ell$ is the index of the spin-weighted spheroidal harmonics, and $n$ and $k$ are the harmonics of the radial and polar motion, respectively. When we set the maximum of $\ell$ to 20, we obtain a relative accuracy of $10^{-5}$ even in the highly eccentric case of $e=0.9$. The accuracy is better for lower eccentricity. Our numerical code is expected to be useful for computing templates of the extreme mass ratio inspirals, which is one of the main targets of the Laser Interferometer Space Antenna (LISA).

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arXiv2009

New Numerical Methods to Evaluate Homogeneous Solutions of the Teukolsky Equation II. Solutions of the Continued Fraction Equation

We investigate the solution of the continued fraction equation by which we determine "the renormalized angular momentum parameter", $ν$, in the formalism developed by Leaver and Mano, Suzuki, and Takasugi. In this formalism, we describe the homogeneous solutions of the radial Teukolsky equation, which is the basic equation of the black hole perturbation formalism. We find that, contrary to the assumption made in previous works, the solution, $ν$, becomes complex valued as $ω$ (the angular frequency) becomes large for each $l$ and $m$ (the degree and order of the spin-weighted spheroidal harmonics). We compare the power radiated by gravitational waves from a particle in a circular orbit in the equatorial plane around a Kerr black hole in two ways, one using the Mano-Suzuki-Takasugi formalism with complex $ν$ and the other using a direct numerical integration method. We find that the two methods produce consistent results. These facts prove the validity of using complex solutions to determine the homogeneous solutions of the Teukolsky equation.

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arXiv2004

New Numerical Methods to Evaluate Homogeneous Solutions of the Teukolsky Equation

We discuss a numerical method to compute the homogeneous solutions of the Teukolsky equation which is the basic equation of the black hole perturbation method. We use the formalism developed by Mano, Suzuki and Takasugi, in which the homogeneous solutions of the radial Teukolsky equation are expressed in terms of two kinds of series of special functions, and the formulas for the asymptotic amplitudes are derived explicitly.Although the application of this method was previously limited to the analytical evaluation of the homogeneous solutions, we find that it is also useful for numerical computation. We also find that so-called "renormalized angular momentum parameter", $ν$, can be found only in the limited region of $ω$ for each $l,m$ if we assume $ν$ is real (here, $ω$ is the angular frequency, and $l$ and $m$ are degree and order of the spin-weighted spheroidal harmonics respectively). We also compute the flux of the gravitational waves induced by a compact star in a circular orbit on the equatorial plane around a rotating black hole. We find that the relative error of the energy flux is about $10^{-14}$ which is much smaller than the one obtained by usual numerical integration methods.

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