Waveforms produced by a particle plunging into a black hole in massive gravity : Excitation of quasibound states and quasinormal modes

Transcription

1 Waveforms produced by a particle plunging into a black hole in massive gravity : Excitation of quasibound states and quasinormal modes Mohamed OULD EL HADJ Université de Corse, Corte, France Projet : COMPA Équipe physique théorique THE ERA OF GRAVITATIONAL WAVE ASTRONOMY Institut d astrophysique de Paris I.A.P) France June 26-30, In collaboration with Yves DÉCANINI and Antoine FOLACCI : [arxiv: ] Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

2 Introduction In the context of Einstein s general relativity, the construction of the waveform produced by a plunging particle is of fundamental importance because : i ii The plunge regime is the last phase of the evolution of a stellar mass object orbiting near a supermassive BH. The waveform generated during this regime encodes the final BH fingerprint. i ii iii With in mind the possibility to test massive gravity in the context of BH physics : We consider the radiation produced by a particle plunging from slightly below the innermost stable circular orbit ISCO) into a Schwarzschild BH. In order to circumvent the difficulties associated with BH perturbation theory in massive gravity, we use, at first, a toy model where we replace the graviton field by a massive scalar field and consider a linear coupling between the particle and this field. We compute the waveform generated by the plunging particle and study its spectral content. We highlight some important effects which are not present for massless fields. iv We compute the waveform generated by the plunging particle for the odd-parity l = 1 mode in massive gravity. Throughout this presentation : i We display our numerical results by using the dimensionless coupling constant α = 2Mµ m 2 p here M, µ and mp denote respectively the mass of BH, the rest mass of the field and the Planck mass). 1) ii We adopt natural units ħ = c = G = 1). Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

3 Our model We consider the exterior of the Schwarzschild BH of mass M M,g αβ ) defined by the metric ds 2 = g µν x)dx µ dx ν = 1 2M ) dt r 1 2M ) dr 2 + r 2 dσ 2 2, 2) r where dσ 2 2 = dθ2 + sin 2 θdϕ 2 denotes the metric on the unit 2 sphere S 2. The particle is coupled to a scalar field φ with mass µ and the dynamics of the system field-particle is defined by the action : S = S field + S particle + S interaction 3) with and with S field = 1 2 d4 x gx) M S particle = m 0 dτ = m 0 γ [ g αβ x) α Φx) β Φx) + µ 2 Φ 2 ] x), 4) γ g αβ zλ)) dzα λ) dλ dz β λ) dλ dλ 5) S interaction = d4 x gx)ρx)φx) 6) M ρx) = q dτδ 4 x,zτ)) charge density. 7) γ here, m 0 mass of the particle, q its scalar charge and z α = z α τ) describe its world line γλ). Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

4 Waveform associated with the l,m) mode Due to both the staticity and the spherical symmetry of the Schwarzschild background, the wave equation reduces to the Regge-Wheeler equation with source [ ] d 2 dr 2 + ω 2 V l r) φ ωlm = 1 2M r with the effective potential V l r) given by V l r) = 1 2M r and the tortoise coordinate r r) is given by )[ µ 2 + ) ρ ωlm. 8) ll + 1) r 2 + 2M ] r 3. 9) r r) = r + 2M ln[r/2m 1]. 10) For r 2M black hole horizon), r r) For r + spatial infinity), r r) + In order to solve the Regge-Wheeler equation, we use the machinery of Green s function. We can show that the solution of Eq 8 is given by φ lm t,r) = 1 + +ic e iωt + dω 2π +ic W l ω) φup ωl r) 2M dr φ in ωl r )ρ ωlm r ). 11) Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

5 IN-Mode and UP-Mode φ in and φup are linearly independent solutions of the homogenous Regge-Wheeler equation ωl ωl d 2 φ [ ωl dr 2 + ω 2 ] V l r) φ ωl = 0. 12) When Imω) > 0, φ in is uniquely defined by its ingoing behavior at the event horizon r = 2M i.e., for ωl r ) φ in ωl r) r e iωr 13a) and, at spatial infinity r + i.e., for r + ), it has an asymptotic behavior of the form [ ] φ in ω 1/2 ) ωl r) r + A ) pω) l ω)e i[pω)r +[Mµ2 /pω)]lnr/m)] +A +) l ω)e+i[pω)r +[Mµ2 /pω)]lnr/m)] 13b) Similarly, φ up is uniquely defined by its outgoing behavior at spatial infinity ωl [ ] φ up ω 1/2 ωl r) r + e +i[pω)r +[Mµ2 /pω)]lnr/m)] 14a) pω) and, at the horizon, it has an asymptotic behavior of the form φ up ωl r) r B ) l ω)e iωr + B +) l ω)e+iωr. 14b) In Eqs. 13) and 14), pω) = ω 2 µ 2) 1/2 denotes the wave number" while A ) ω), A+) ω), B ) ω) and l l l B +) ω) are complex amplitudes. l By evaluating the Wronskian W l ω) at r and r +, we obtain W l ω) = 2iωA ) ω) = 2iωB+) ω). 15) l l Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

6 Source due to the particle on a plunge trajectory We consider a particle plunging into the BH from r ISCO = 6M Innermost Stable Circular Orbit). The plunge trajectory lies in the equatorial plane θ = π/2) is given by t p r) 2M = 2 2r 24M) 2M 6M/r 1) 1/2 22 2tan 1 [6M/r 1) 1/2] and +2tanh 1[ 3M/r 1/2) 1/2] + t 0 16) 2 3r ϕr) = 6M r) 1/2 + ϕ 0 17) which can be rewritten FIGURE 1 The plunge trajectory. Here, we assume that the 6M rϕ) = [ /ϕ ϕ0 ) 2]. 18) particle starts at r = r ISCO 1 ɛ) with r ISCO = 6M and ɛ = We have for the source of the plunging particle ρ ωlm r) = 3q r e i [ ] ωtpr) mϕpr) 2π 6M r) 3/2 Y lm π ) 2,0. 19) Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

7 Quadrupolar waveforms produced by the plunging particle Re[ϕ l mt,r)]/ Re[ϕ l mt,r)]/ Re[ϕ l mt,r)]/ l=2, m= α = 0 and r = 10M a) l=2, m= α = 0 and r = 20M b) l=2, m=2 α = 0 and r = 50M c) t/2m) FIGURE 2 Quadrupolar waveforms produced by the plunging particle.the results are obtained for massless scalar field α = 0 ) and for massive scalar field α = 0.25,0.35). The observer is located at a) r = 10M, b) r = 20M et c) r = 50M. Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

8 Quadrupolar waveforms produced by the plunging particle Re[ϕ l mt,r)]/ l=2, m=2 α = 0 and r = 10M a) l=2, m=2 α = 0 and r = 20M Re[ϕ l mt,r)]/ Re[ϕ l mt,r)]/ Adiabatic phase Ringdown phase Late-time tail b) l=2, m=2 α = 0 and r = 50M c) t/2m) FIGURE 2 Quadrupolar waveforms produced by the plunging particle.the results are obtained for massless scalar field α = 0 ) and for massive scalar field α = 0.25,0.35). The observer is located at a) r = 10M, b) r = 20M et c) r = 50M. For the massless scalar field α = 0), the waveforms can be decomposed in 3 phases : Adiabatic phase A ringdown phase A late-time tail For the massive scalar field α 0) : Such a decomposition remains roughly valid for low masses. For a given distance r, the waveform amplitude decreases as the mass increases. Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

9 The adiabatic phase and the circular motion of the particle on the ISCO Particle in circular orbit on the ISCO : The trajectory is given by ϕ p t) = Ω ISCO t où Ω ISCO = 1 6 6M The source is given by πq ρ ωlm r)= 6M δr 6M)δω mω ISCO )Y π2,0). lm and the waveform in l,m) mode is given by q φ lm t,r)= 12 2iM Y π2,0) φup ωl r)φin ωl 6M) lm ωa ) e iωt l ω) FIGURE 3 Comparaison of the quadrupolar waveform produced ω=mωisco by the plunging particle blue line) and by a particle orbiting the BH on the ISCO red dashed line). The results are obtained for a massive scalar field α = 0.25). The adiabatic phase is described very accurately by the waveform emitted by a particle living on the ISCO. The study of the particle orbiting the BH on the ISCO, permits us to define two regimes : The dispersive regime. The evanescent regime. These two regimes are separated by the threshold value α c corresponding to the mass parameter µ c µc=2ω ISCO where Ω ISCO denotes the angular velocity of the particle moving on the ISCO. Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

10 The adiabatic phase and the excitation of QBSs FIGURE 4 Spectral content of the adiabatic phase of the quadrupolar waveform produced by the plunging particle. The results are obtained for a massive scalar field α = 0.25 et 0.35) and the observer is located at a) r = 20M et b) r = 50M. We observe the signature of the quasi-circular motion of the plunging particle. In the evanescent regime : We can observe the signature of the quasi-circular motion. It is important to remark the presence of another peak at the frequency equals to the real part of the complex frequency of the first QBS. In other term we can observe the excitation of the first QBS in adiabatic. Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

11 The ringdown phase and the excitation of QNMs FIGURE 5 Comparaison of the quadrupolar waveform produced by the plunging particle blue line) and the quadrupolar quasinormal waveform red dashed line). The results are obtained for an observer at left) 10M and right) 50M. When α and the distance r are not too large, the quasinormal waveform describes accurately the ringdown phase. However, if α or the distance r increase, the agreement is not so good. Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

12 The late-time phase and the excitation of QBSs FIGURE 6 Spectral content of the late-time phase of the quadrupolar waveform produced by the plunging particle. The results are obtained for a massive scalar field α = 0.25 and 0.35) and the observer is located at a) r = 20M and b) r = 50M. We observe the signature of the long-lived QBS. We note that, as the reduced mass parameter α increases, the spectrum of the frequencies of the QBSs spreads more and more and it is the possible to separate the different excitation frequencies. Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

13 Waveforms in massive gravity Our study will be limited to the Fierz-Pauli theory in the Schwarzschild spacetime which can be obtained by linearization of the ghost-free bimetric theory. The gravitational wave equation The linearized Fierz-Pauli equation) is given by h µν + 2R µρνσ h ρσ µ 2 h µν = 16π τ µν 1 3 g µντ ρ ρ + 1 ), 20) 3µ 2 µ ν τ ρ ρ ) µ 16π h µν = ν 3µ 2 τρ ρ, 21) h = 16π τ. 22) 3µ 2 The field h µν t,r,θ,ϕ) describing the gravitational waves propagating in the Schwarzschild spacetime can be searched h µν t,r,θ,ϕ) = h e) µν t,r,θ,ϕ) + ho) µν t,r,θ,ϕ) Convention : h lm = 0 for l = 1. The general form of the stress-energy tensor source of the gravitational perturbations of the Schwarzschild black hole is given by τ µν t,r,θ,ϕ) = τ e) µν t,r,θ,ϕ) + τo) µν t,r,θ,ϕ) Conventions : Lt lm l = 0,1. = L lm r = 0 for l = 0 and L lm = 0 for 0 0 h lm t X lm h lm θ t X lm ϕ h o) + m=+l µν = sym 0 h lm r X lm h lm θ r X lm ϕ l=1 sym sym h m= l lm X lm h lm X lm θθ θϕ sym sym sym h lm Xϕϕ lm 0 0 L lm t X lm L lm θ t X lm ϕ τ o) + m=+l µν = sym 0 L lm r X lm L lm θ r X lm ϕ l=0 sym sym L m= l lm X lm L lm X lm θθ θϕ sym sym sym L lm Xϕϕ lm Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

14 Waveform in massive gravity odd-parity) The system of coupled equations governing the oddparity modes [ ] d 2 dr 2 +ω 2 V φ) l r) φ ωlm + l 1)l+2) r 2 1 2M ) 1 3M ) ψ r r ωlm = S φ) ωlm [ ] d 2 dr 2 +ω 2 V ψ) l r) ψ ωlm + 4 r 2 1 2M ) φ r ωlm = S ψ) ωlm With φ ωlm and ψ ωlm φ ωlm = 1 2M ) h lm r r ψ ωlm = hlm r The effectives potentials are given by V φ) 1 l r) = 2M ) r µ 2 + ll+1)+4 r 2 V ψ) 1 l r) = 2M ) r µ 2 + l 1)l+2) r 2 16M ) r 3 + 2M ) r 3 The source terms are given by S φ) 16π 1 2M ) 2 ωlm = r + dtllm r e +iωt 2π S ψ) 16π 1 2M ) ωlm = r + dtllm e +iωt 2πr In the case of the odd-parity l = 1 mode : the system of equation reduces to a single differential equation Regge-Wheeler type [ d 2 ] dr 2 + ω 2 V φ) l r) φ ωlm = S φ) ωlm Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

15 Excitation of the odd-parity l = 1 mode by the plunging particle the source of the plunging particle odd-parity) is given by S φ) ωlm = 16 6π Mm 0 ll+1) r 2 1 2M ) Bl,m) e i[ ωtpr) mϕpr) ] 23) r with Bl,m) = 2m+1 2l+1 π 4π l m)! Γ l+m) 2 l+m)! Γ ) +1 l m 1) 2 +1 [ π ] ) sin 2 l+m) We can show that the partial response in the odd-parity l = 1 mode is given by φ lm t,r) = 1 ) + +ic e iωt dω φ up + 2π +ic W l ω) ωl r) 2M dr φ in ωl r )S φ) ωlm r ). 25) 24) Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

16 Excitation of the odd-parity l = 1 mode by the plunging particle Waveforms for the odd-parity l = 1 mode Re[ϕ l m t,r)]/m Fierz-Pauli field odd-parity sector) l=1, m=0 α 0 and r = 50M Re[ϕ l m t,r)]/m FT[ϕ lm adiabatic t,r)]/ QBS 2Mω Fierz-Pauli field odd-parity sector) l=1, m=0 FT[ϕ lm Late-time t,r)]/ QBS 2Mω α = 0.15 and r = 50M t/2m) t/2m) Re[ϕ l m t,r)]/m FT[ϕ lm adiabatic t,r)]/ 2Mω QBS Fierz-Pauli field odd-parity sector) l=1, m=0 α = 0.25 and r = 50M FT[ϕ lm Late-time t,r)]/ QBS 2Mω Re[ϕ l m t,r)]/m FT[ϕ lm adiabatic t,r)]/ 2Mω QBS Fierz-Pauli field odd-parity sector) l=1, m=0 α = 0.89 and r = 50M FT[ϕ lm Late-time t,r)]/ QBS 2Mω t/2m) t/2m) FIGURE 7 Waveforms l = 1,m = 0) produced by the plunging particle.the results are obtained for massless scalar field α 0) and for massive field α = 0.15,0.25 and 0.89). The observer is located at r = 10M. We can observe for any nonvanishing value of the reduce masse parameter α, the QBSs of the Schwarzschild BH are excited, their influence is negligible for α 0 but increases with α. Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

17 Conclusion and perspectives I) Summary : In our opinion, the study of the scalar radiation generated by the plunging particle has permitted us to highlight and interpret some important effects occurring in the plunge regime which are not present for massless fields such as : i The decreasing and vanishing, as the mass parameter increases, of the signal amplitude generated when the particle moves on quasicircular orbits near the ISCO. ii In addition to the excitation of the QNMs, the excitation of the QBSs of the BH. For the massive gravity, we have restricted our study to the odd-parity l = 1 partial mode of the Fierz- Pauli theory in the Schwarzschild spacetime and its behavior is governed by a single differential equation of the Regge-Wheeler type while all the other partial modes are governed by two or three coupled differential equations depending on the parity sector and the angular momentum. II) Perspectives : it would be interesting to consider partial modes with higher angular momentum l 2). It would be important to extend our study to a rotating BH in massive gravity : because, if the graviton has a mass, the study of the influence of the QBSs on the gravitational radiation emitted by a rotating BH and, in particular, the observation of the effects previously discussed, could help us : i To test the various massive gravity theories. ii To impose strong constraints on the graviton mass and to support, in a new way, Einstein s general relativity. Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16

18 Conclusion and perspectives I) Summary : In our opinion, the study of the scalar radiation generated by the plunging particle has permitted us to highlight and interpret some important effects occurring in the plunge regime which are not present for massless fields such as : i The decreasing and vanishing, as the mass parameter increases, of the signal amplitude generated when the particle moves on quasicircular orbits near the ISCO. ii In addition to the excitation of the QNMs, the excitation of the QBSs of the BH. For the massive gravity, we have restricted our study to the odd-parity l = 1 partial mode of the Fierz- Pauli theory in the Schwarzschild spacetime and its behavior is governed by a single differential equation of the Regge-Wheeler type while all the other partial modes are governed by two or three coupled differential equations depending on the parity sector and the angular momentum. II) Perspectives : it would be interesting to consider partial modes with higher angular momentum l 2). It would be important to extend our study to a rotating BH in massive gravity : because, if the graviton has a mass, the study of the influence of the QBSs on the gravitational radiation emitted by a rotating BH and, in particular, the observation of the effects previously discussed, could help us : i To test the various massive gravity theories. ii To impose strong constraints on the graviton mass and to support, in a new way, Einstein s general relativity. Thank you for your attention. Particle plunging into a black hole in massive gravity : Excitation of... TEGRAW I.A.P June / 16