Probing the Ultralight Boson with Black Hole Ringdown
arXiv:2107.05492
Joseph Gais1, Adrian Ka-Wai Chung2, Mark Ho-Yeuk Cheung1 3, Tjonnie Guang Feng Li1
1The Chinese University of Hong Kong
2King’s College London
3John Hopkins university
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Ultralight Bosons and Cloud Formation
Ultralight boson: dark matter candidate, scalar1 Mass1 eV
Bosonic cloud formation around black holes due to superradiant instabilities2
For massive boson → quasi-bound state3
“Gravitational atom”
1David JE Marsh. “Axion cosmology”. In: Physics Reports643 (2016), pp. 1–79.
2Richard Brito, Vitor Cardoso, and Paolo Pani. “Superradiance”. In: Lect. Notes Phys906.1 (2015), pp. 1501–06570.
3Daniel Baumann, Horng Sheng Chia, and Rafael A Porto. “Probing ultralight bosons with binary black holes”. In: Physical Review D99.4 (2019), p. 044001.
How can we probe ultralight bosons?
Direct measurement of monochromatic radiationa
Stochastic backgroundb Dynamical frictionc
“Holes” in Regge-Wheeler planed What about ringdown signatures?
aMaximiliano Isi et al.“Directed searches for gravitational waves from ultralight bosons”. In: Physical Review D99.8 (2019), p. 084042.
bLeo Tsukada et al.“First search for a stochastic gravitational-wave background from ultralight bosons”. In: Physical Review D99.10 (2019), p. 103015.
cMiguel C Ferreira, Caio FB Macedo, and Vitor Cardoso. “Orbital
fingerprints of ultralight scalar fields around black holes”. In:Physical Review D 96.8 (2017), p. 083017.
dRichard Brito, Vitor Cardoso, and Paolo Pani. “Superradiance”.In: Lect.
Notes Phys906.1 (2015), pp. 1501–06570,Ken KY Ng et al.“Constraints on Ultralight Scalar Bosons within Black Hole Spin Measurements from the LIGO-Virgo GWTC-2”. In: Physical Review Letters126.15 (2021), p. 151102.
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Bosonic cloud wavefunction
Bosonic cloud wavefunction for l =m= 1 mode ϕ(t,r, θ, φ) =
r 3Ms
4πIM(Mµ)3µre−Mµ2r/2sinθcos (φ−µt), (1) I: O(1) factor
M: BH mass
Ms: Total mass of bosonic cloud µ= ms
~
Energy-momentum tensor given by Tµν =−gµν
1
2gαβ∂αΦ∂βΦ + 1 2µ2Φ2
+∂µΦ∂νΦ (2)
Teukolsky Equation
Teukolsky equation for gravitational perturbations
JΨ = 4πΣT (3)
J differential operator ofr, θ, φ,t
Energy-momentum tensor from bosonic cloud contributes to source term
How does it affect the ringdown waveform?
∆ =r2−2Mr +a2 ρ=r−iacosθ
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Source term in Teukolsky Equation
Relevant source term in Teukolsky equation
Tm¯m¯ = ¯mβm¯γTβγ (4)
¯
mµ= 1 r√
2(0,0,1,−isin−1θ) (5) Arrive at
Tm¯m¯ =−1
2m¯βm¯γ
hβγ(∂αΦ∂αΦ +µ2Φ2)
(6) h¨βγ ≈ −ω2hβγ (7) and thus
Tm¯m¯ ∝ψ4 (8)
ψ4(r → ∞) = 12
h¨+−i¨h×
Source term acts as effective potential
From Potentials to Reflectivity
Radial equation decoupled: modified wave equation with extra from source term
∂2R
∂x2 + (ω2−V(r)−VULB(r))R(r, ω) = 0 (9) V(r) usual potential in Teukolsky equation
VULB extra contribution from bosonic cloud How does this potential affect waveforms?
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Bosonic Cloud Potential
0 200 400 600 800 1000
r/M 0.0
0.2 0.4 0.6 0.8 1.0
VULB(r)
×10−5
Thepeakof|Φ(t,r,θ,φ)|2
µ= 0.1, M= 1, a= 0.4, Ms= 0.05, ω= ˜ω022
VULBRe VULBIm
Peak ofVULB atrpeak ∝ Mµ12
From Potentials to Quasinormal-Mode frequencies
Bosonic cloud→ ”dirty” black hole VULB(r)∝(Mµ)8 1
Treat bosonic cloud as perturbing potential of Schwarzschild background
Logarithmic perturbation theory for perturbative potential4 Calculate QNM shifts froml =m= 1 cloud mode on nlm= (022,021) GW modes
ω˜n`m ≈ω˜n`m(0) +∆n`m(µ,Ms,M)
2˜ω(0)n`m (10)
4PT Leung et al.“Quasinormal modes of dirty black holes”.In: Physical review letters78.15 (1997), p. 2894.
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Quasinormal-Mode Shifts
For Mµ <0.5,ω022Im decreases, ω022Re increases
0.44 0.46 0.48
M ωRe022
−0.16
−0.14
−0.12
−0.10
−0.08
Mω
Im 022
−17.00
−16.95
−16.90
−16.85
−16.80
−16.75
−16.70
−16.65
−16.60
log10µ(eV)
Waveform with Bosonic Cloud
Frequency domain waveform model
h(f;Mf,af, µ,Ms,An`m, φn`m)
=−2m r
X
n`m
An`mωIm,n`me−iφn`m
ωIm,n`m2 + (2πf −ωRe,n`m2 )2. (11) Likelihood of signal s with parameterized waveformh(~θ)
L(s|~θ)∝exp
−1
2hs−h(~θ)|s −h(~θ)i
(12) Conduct injection for EMRI at Sag A and M32 as proof-of-principle detected by LISA with noise
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Results for Sag A
−18.00 −17.75 −17.50 −17.25 −17.00 −16.75 −16.50 −16.25 −16.00
log10µ [eV]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
p(log10µ[eV]|d,H,I) 90-CIof logµbySgrA* µ=M−1 SgrA∗
Sagittarius A*
M32, assumeda= 0.9
Results of µ > 0
−17.075−17.050−17.025−17.000−16.975−16.950−16.925 log10µ[eV]
0.0 2.5 5.0 7.5 10.0 12.5 15.0
p(log10µ[eV]|d,H,I)
Injectedµ Sgr A*
(a)Sagittarius A*
−18.0 −17.8 −17.6 −17.4 −17.2 −17.0 −16.8 log10µ[eV]
0 1 2 3 4 5 6
p(log10µ[eV]|d,H,I)
Injectedµ M32,a= 0.9
(b)M32
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Possibilities for better rates
Main limitation: ringdown SNR scales as q = (m/M)2
EMRI ringdown SNR very small relative to equal mass binaries But inspiral analysis suggests cloud depletes at equal mass ratios
Figure:Possible for cloud to survive even for q∼0.15
5Emanuele Berti et al.“Ultralight boson cloud depletion in binary systems”.In:
Physical Review D99.10 (2019), p. 104039.
Takeaways
l =m= 1 bosonic cloud leaves imprint on QNM frequencies of dominant GW mode, detectable in nearby EMRIs
QNM shift effect in IMBH-SMBH mergers could be detectable at larger distances
Method could possibly extend to other matter sources
Gold standard: IMR consistency test of boson mass measurements, akin to work on consistency of inspiral methods6
6Otto A Hannuksela et al.“Probing the existence of ultralight bosons with a single gravitational-wave measurement”. In: Nature Astronomy3.5 (2019), pp. 447–451.
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The End
Figure: Link to our manuscript on arXiv: 2107.05492
Eigenfunctions ψ4=ρ4X
l,m
Z d˜ω
√2πSnlm(θ, φ)e−iωt˜ Rlm˜ω(r) (13)
l =m= 2 mode of spin-weighted spheroidal harmonic Extract Teukolsky source term
T˜(t,r,Ω)∼ −1
4ρ8ρ∆¯ 2Jˆ+[ρ−4Jˆ+(ρ−2ρT¯ m¯m¯)] (14) Jˆ+=∂r −∆−1(r2∂t)
R(r, ω)∝eiωr
Ansatz: ∂R∂r ∼iωR(r)
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