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Quasinormal Ringing of Acoustic Black Holes 京都大学大学院 人間・環境学研究科 宇宙論・重力グループ M2 奥住 聡 共同研究者：阪上雅昭（京大 人・環）, 吉田英生（京大 工）

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What is an Acoustic Black Hole? “Acoustic BH” = Transonic Flow downup sonic point “effective” sound velocity in the lab Acoustic BH region In the supersonic region, sound waves cannot propagate against the flow. = sonic horizon “Acoustic Black Hole”

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throat “Laval Nozzle”: Convergent-Divergent Nozzle

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THEORY Graduate School of H&E Studies EXPERIMENT Graduate School of Engineering TARGETS Hawking Radiation Quasinormal Ringing numerical Planckian fit Acoustic BH Experiment Project at Kyoto Univ.

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compressor mass flow meter settling chamber Laval nozzle flow 20cm Configuration

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Quasinormal Ringing “Characteristic ‘sound’ of BHs (and NSs)” occurs when the geometry around a BH is perturbed and settles down into its stationary state. e.g. after BH formation / test particle infall Described as a superposition of a countably infinite number of damped sinusoids (QuasiNormal Modes, QNMs). QNM frequencies contain the information on (M,J) of BHs.

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Quasinormal Ringing of a BH NS-NS merger to a BH (Shibata & Taniguchi, 2006) QN ringing inspiral phasemerger phase

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Definition of QNMs Schroedinger-type wave eq. outgoing BCswith QNMs are defined as solutions of horizoninfinity solutions: n : complex

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Example: Schwarzschild Black Hole

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horizonspatial infinity Example: Schwarzschild Black Hole ~ r g -2

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Example: Schwarzschild Black Hole fundamental (n=0) mode

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Goal of This Study What form of nozzle yield the QNM easiest to observe? A measure of the detectability: “Q-value” Are QNMs excited in experimentally feasible situations? Numerical simulation (full order calculation) Find the form of the nozzle which yield large Q Experimental testing of gravitational-wave analysis (such as matched filtering) Does acoustic BH really have QNMs? Questions

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-- wave eq. for velocity potential perturbation Sound Waves in Inhomogeneous Fluid Flow Perturbation: 1D flow approx.:

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Schroedinger-type Wave Equation c s0 : sound speed at stagnation points , v, A : independent of t M : Mach number where “effective potential”

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Potential Barrier for Different Laval Nozzles Consider a family of Laval nozzle: nozzle radius K : integer rr tank 1tank 2nozzle flow

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Potential Barrier for Different Laval Nozzles 1.04L 3.92L L L flow sonic horizon flow sonic horizon

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Procedure for Calculating QN Freq’s Calculate the “S-matrix” for the potential barrier V( ): Then, impose the outgoing B.C., and find ’s that satisfy the boundary condition. : “S-matrix”

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WKB Approach 00 Region (I) & (III): WKB solutions for truncated V( ) Around : exact solution for truncated V( ) Expand V( ) in a Taylor series about the maximum point 0 : (I ) (II)(III ) 1 st order: Schutz & Will, rd order: Iyer & Will, th order: Konoplya, 2004 Matching matching regions 23 12

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WKB Approach: S-Matrix Here, is related to by where (1 st WKB)

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QNM Solutions by WKB Approach Conditions for QNMs: i.e. QNM frequency (1 st WKB value)

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QNM Frequencies of Different Laval Nozzles (WKB approx. breaks down) Q = 2 Q = 1 WKB solution: Schutz & Will (1985) Iyer & Will (1987) : peak point of V n=0 mode freq. (3 rd WKB value)

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Numerical Simulation of Acoustic QN Ringing We perform two types of simulations: “Acoustic BH Formation” initial state: no flow set sufficiently large pressure difference final state: transonic flow “Weak Shock Infall” initial state: transonic flow let a weak shock “fall” into the horizon final state: transonic flow ~ BH formation~ test particle infall flow

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Example of Transonic Flow flow sonic horizon supersonicsubsonic

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Result 1: Weak Shock Infall steady shock horizon gif weak shock

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Result 1: Weak Shock Infall steady shock horizon weak shock QN ringing gif

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QNM fit (3 rd WKB) numerical ringdown phase observed waveform Result 1: Weak Shock Infall

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QNM fit (3 rd WKB) numerical nonlinear phaseringdown phase Result 2: Acoustic BH Formation observed waveform

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QNM fit (3 rd WKB) numerical nonlinear phaseringdown phase Result 2: Acoustic BH Formation observed waveform Numerical waveform agrees with the least damped QNM very well!

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Numerical Simulation: Discussion In both types of simulations, QNMs are actually excited. The results agree with WKB analysis well ( for K >1 ). Typical values in laboratories: cf. real BH: ( l=m=2, least- damped ) >2.0 QNMs of acoustic BHs decay too quickly. Difficult to detect in experiments…?

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Partially Reflected Quasinormal Modes (PRQNMs) outgoing B.C. + “half mirror” B.C. “half mirror” cc

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Example: Contact Surface in Perfect Fluid Contact surface (contact discontinuity): discontinuity of the density . the pressure p and the fluid velocity v are continuous. moves with the surrounding fluid, i.e., v c = v. partially reflects sound waves. vcvc vv 12 Contact Surface (C.S.)

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Example: Contact Surface in Perfect Fluid vcvc vv 12 If v c (= v) << c s, refl. coeff. R( ) for sound waves propagating from 1 to 2 is given by [e.g. Landau & Lifshitz, Fluid Mechanics] C.S.

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PRQNM Solutions by WKB Approach Partially Reflecting B.C. :

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Example: Contact Surface in Perfect Fluid Re Im Table: the least damped PRQNM frequency (3 rd WKB value) contact surface enhances Q-value!!

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observed waveform PRQNM fit (3 rd WKB) numerical Numerical Simulation of PRQNMs

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no contact surfacecontact surface present

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Summary For future experiments, we have studied QN ringing of acoustic BHs in Laval nozzles. A contact surface elongates the damping times of QNMs. Acoustic BHs (transonic fluid flow) do have QNMs. QNMs are excited in experimentally feasible situations. A wider range of Q becomes accessible! Experimental testing of gravitational-wave analysis (such as matched filtering). Astrophysical BH surrounded by a “half mirror” ?? Future Works

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PRQNM Solutions by WKB Approach In region (III), right-going WKB sol.left-going WKB sol. cc region (III)region (IV) 23

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PRQNM Solutions by WKB Approach In region (III), right-going WKB sol.left-going WKB sol. Furthermore, if c lies far away from the potential barrier,

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QNM fit PRQNM fit Numerical Simulation of PRQNMs For t <15, an “ordinary” QNM (not PRQNM) dominates, since the potential barrier is not yet “aware” of the contact surface.

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