Gravitational collider physics uses binary black hole mergers to probe ultralight bosons through their effect on gravitational waves. Bound boson fields form "gravitational atoms" around black holes that can undergo transitions when perturbed by an orbiting companion. This leads to distinctive signatures in the gravitational waves like floating or sinking orbits and sharp features in the frequency evolution. Ionization of the cloud can also significantly shorten the merger time. Precise waveform modeling is still needed to apply these effects in gravitational wave searches.

1. Gravitational Collider Physics
Daniel Baumann

2. Based on work with
H.S. Chia R. Porto J. Stout G. Tomaselli G. Bertone
DB, Chia and Porto
Probing Ultralight Bosons with Binary Black Holes
arXiv:1804.03208 PRD [Editor’s Suggestion]
DB, Chia, Porto and Stout
Gravitational Collider Physics
arXiv:1912.04932
DB, Bertone, Stout and Tomaselli
Ionization of Gravitational Atoms
arXiv:2112.14777
DB, Bertone, Stout and Tomaselli
Sharp Signals of Boson Clouds in Black Hole Binary Inspirals
arXiv:22xx.xxxxx (PRL accepted) [Editor’s Suggestion]
DB, Chia, Stout and ter Haar
The Spectra of Gravitational Atoms
arXiv:1908.10370

3. Can it also become a new tool for fundamental physics?
The detection of GWs marked the beginning of a new era of multi-messenger
astronomy:

4. particle collider
Standard
Model
Neutrinos
1020
1010
1
10 10
10 20
Energy [eV]
New physics is often associated with high energies (short distances):
We expect this to have small effects on long-wavelength GW signals.
cosmological
collider

5. New physics may also occur at low energies, if it is weakly coupled:
particle collider
Standard
Model
Neutrinos
1020
1010
1
10 10
10 20
Energy [eV]
EM
Strong
Weak
Gravity
Coupling
gravitational
collider
This may be probed by GW signals from binary black hole mergers.
cosmological
collider

6. superradiance
Ultralight bosons can form bound states around rotating black holes:
This resembles the hydrogen atom and is therefore often called a
gravitational atom.
Zel’dovich ’72
Starobinsky ’73
Arvanitaki et al ‘09
superradiance

7. A companion will perturb the cloud with a slowly increasing frequency.
DB, Chia, Porto, and Stout ’19

8. DB, Chia, Porto, and Stout ’19
This interaction is enhanced at certain resonance frequencies.

9. DB, Chia, Porto, and Stout ’19
The backreaction of the cloud’s dynamics on the orbit leaves distinct
imprints in the gravitational waves emitted by the binary.

10. Outline
1) Gravitational Atoms
2) Evolution in Binaries
3) Gravitational Wave Signals

11. GRAVITATIONAL
ATOMS
I.

12. A scalar field around a rotating black hole is described by the KG equation:
g↵
r↵r µ2
= 0
We will focus on fields whose Compton wavelength is larger than the BH:
Ultralight Bosons around Black Holes
1 km 1010
km
10 10
eV 10 20
eV

13. Such light fields see the far-field limit of the geometry:
Ultralight Bosons around Black Holes
µ2
ds2
=
✓
1
2↵
r
◆
dt2
+
✓
1 +
2↵
r
◆
dr2 4↵2
ã sin2
✓
r
dt d + r2
d⌦2
where ↵ ⌘ rg/ c is the gravitational fine-structure constant.
spin
i
d
dt
=
✓
1
2µ
r2 ↵
r
+ V
◆
The KG equation then reduces to a Schrödinger equation:
with = (t, x)eiµt
Coulomb
potential
fine structure
corrections
V

14. 0 2 6 12
αr
|ψ
n!m
(r)|
2
|10m!
|21m!
|32m!
|43m!
Bound States
The energy eigenstates are the same as those of the hydrogen atom:
with “quantum numbers”:
principal orbital azimuthal
m
`
n
r/r0

15. 0 5 10
µαr
|ψ
k;!m
(r)|
2
|0; 11!
|1; 11!
|2; 11!
|3; 11!
|211!
Unbound States
There are also unbound continuum states:
r/r0
with “quantum numbers”:
energy orbital azimuthal
m
`
E

16. Bound State Spectrum
En`m = µ
✓
1
↵2
2n2
+ fn`↵4
+ hn`m↵5
+ · · ·
◆
The bound state spectrum is the same as for the hydrogen atom:
! = 0
|211!
|311!
! = 1
|322!
! = 2
n = 1
n = 2
n = 3

17. ` = 0
|211i
|311i
` = 1
|322i
` = 2
n = 1
n = 2
n = 3
Growing and Decaying Modes
Unlike in the hydrogen atom, the states are only quasi-stationary:
!n`m = En`m + i n`m
decaying
growing

18. ΩH
ω < mΩH
Superradiant Scattering
The growing modes are populated by superradiance:
Outgoing wave
Incoming wave
/ exp(im i!t)
This is the wave analog of the Penrose process.
Penrose ’71
Zel’dovich ’72
Starobinsky ’73

19. Black Hole Bomb
A reflecting mirror around the BH creates a black hole bomb:
Press and Teukolsky ‘72

20. For a massive field, the mirror is provided by the angular momentum barrier:
This creates an exponential amplification of small fluctuations.
Superradiant Instability

21. Growth Rates
1013
1011
109
107
105
103
10
0.02 0.05 0.1 0.15 0.2 0.25
α
|211!
|322!
|321!
The different bound states grow at different rates: n`m / ↵4`+5
Detweiler ‘80

22. Initial State of the Cloud
adiance
The fastest growing mode is the state
We take this as the initial state of the cloud.
1
211 = 103
yrs
✓
M
60M
◆ ✓
0.04
↵
◆9
|211i:

23. T211 = 1010
yrs
✓
M
60M
◆ ✓
0.04
↵
◆15
Lifetime of the Cloud
adiance
In isolation, the gravitational atom is very long lived:
We would like to understand what happens to the atom when it is part of a
binary system.

24. EVOLUTION IN
BINARIES
II.

25. The binary companion can induce transitions between bound states
(“Rabi oscillations”) and excite unbound states (“photoelectric effect”):
DB, Chia and Porto 2018
DB, Chia, Porto and Stout 2019
DB, Bertone, Stout and Tomaselli 2021

26. We will first study the transitions between bound states:
Ω = ∆E
∆E
DB, Chia and Porto 2018
DB, Chia, Porto and Stout 2019

27. Gravitational Perturbation
ϕ∗
M∗
R∗
In a binary, the cloud gets perturbed at a slowly increasing frequency:
i
d
dt
=
✓
1
2µ
r2 ↵
r
+ V + V⇤(R⇤(t), '⇤(t))
◆
This perturbation induces level mixing:
ha|V⇤(t)|bi = ⌘ab(t)e i(ma mb)'⇤(t)

28. ⌦i =
Ei
mi
⇠ 0.01 Hz
✓
60M
M
◆ ⇣ ↵
0.04
⌘3
Resonant Transitions
At specific frequencies, the perturbation is resonantly enhanced:
⌦1
⌦2
⌦3

29. Selection Rules
! = 0
|211!
∆E
|311!
|31 −1!
! = 1
|322!
! = 2
n = 1
n = 2
n = 3
For a scalar atom, the level mixings typically involve only two states.
Only certain transitions are allowed by selection rules:

30. Ω = ∆E
ϕ∗(t)
ηeiϕ∗(t) ηe−iϕ∗(t)
∆E
Two-State System
A simple two-state system is therefore a good model for the dynamics:
H =
1
2 E ⌘ei'⇤(t)
⌘e i'⇤(t) 1
2 E
!

31. HD =
(⌦(t) E)/2 ⌘
⌘ (⌦(t) E)/2
!
Two-State System
Rotating along with the companion, isolates the slow motion:
E

32. Near the resonance, we can write ⌦(t) = ⌦res + t
determines nature of transition
HD(t) =
t
2
✓
1 0
0 1
◆
+ ⌘
✓
0 1
1 0
◆
Landau ’32
Zener ’32
E
Two-State System

33. Landau-Zener Transition
Landau ’32
Zener ’32
The analogous problem in QM was solved by Landau and Zener.
The evolution depends on the size of the parameter :
z ⌘ ⌘2
/
40 20 0 20 40
0
0.5
1
z = 102
p
t
|c
a
(t)|
2
40 20 0 20 40
0
0.5
1
z = 10 2
p
t
|c0|2
|c1|2
• For slow transitions, the initial state gets fully converted.
• For fast transitions, some of the initial state can survive.

34. Multi-State Transitions
Clouds of spinning particles involve multi-state transitions:
50 0 50 100 150
0
0.5
1
p
t
|c
a
(t)|
2
|c1|2
|c2|2
|c3|2
50 0 50
p
t
• The final state is generically a superposition of states.
• There can be neutrino-like oscillations between the states.

35. Backreaction on the Orbit
The transitions force the cloud’s energy and angular momentum to change:
tbefore tres tafter
Sc(t)
Sc(t)
tres
These changes must be balanced by changes in the orbital dynamics.

36. Backreaction on the Orbit
Conservation of angular momentum requires
d
dt
⇣
L + Sc
⌘
= TGW ⇡ 0
cloud
orbit
The dynamics of the cloud affects the orbital motion!
DB, Chia, Porto, and Stout ’20

37. 1 0.5 0 0.5 1 1.5 2
0
0.5
1
1.5
2
t/⌦r
⌦/⌦
r
1 0.5 0 0.5 1
t/⌦r
Floating Orbits
The binary floats when it absorbs angular momentum from the cloud:
The binary emits transient, nearly monochromatic GWs.
DB, Chia, Porto, and Stout ’20

38. 1 0.5 0 0.5 1 1.5 2
0
0.5
1
1.5
2
t/⌦r
⌦/⌦
r
1 0.5 0 0.5 1
t/⌦r
Sinking Orbits
The binary sinks when it releases angular momentum to the cloud:
This leads to a dephasing in the GW signal.
DB, Chia, Porto, and Stout ’20

39. Multiple Transitions
The sequence of transitions is a fingerprint of the atomic spectra:
⌦1
⌦2
⌦3
t
⌦(t)
⌦1
⌦2
⌦3

40. Multiple Transitions
The evolution can be described by a series of scattering events:
⌦1
⌦2
⌦3
⌦(t) = ⌦1 ⌦(t) = ⌦2 ⌦(t) = ⌦3
t
| i S1| i S2S1| i S3S2S1| i
DB, Chia, Porto, and Stout ’20

41. Final State of the Cloud
Qc(t) =
X
a
|ca(t)|2
Qc,a +
X
a6=b
|ca(t)cb(t)| Qab cos( Eab t)
scalar cloud vector cloud
The S-matrix formalism predicts the final state of the cloud:
|f i =
N
Y
n=1
Sn | i i
The shape and time dependence of this state depends on the internal
structure of the cloud:

42. E
Gravitational Collider Physics
• Mass: Position of the resonance
• Spin: Angular dependence
• Mass: Position of the resonances
• Spin: Structure of the final state
As in ordinary particle colliders, the signals depend on the mass and spin of
the particles:

43. 10 -3
10 -2
10 -1
1 2
fres (Hz)
10 -1
1
10
102
103
104
M
§
/M
1
0
-
6
1
0
-
4
1
0
-
2
1
1
0
2
1%
10%
100%
M = 60MØ
0.01 0.05 0.1 0.2 0.3
Æ
10 -8
10 -6
10 -4
10 -2
1
102
104
¢t
tot
(yrs)
Gravitational Wave Signatures
• The resonances naturally occur in the LISA band:
• The dephasing of the signals can be significant:

44. III. SIGNATURES OF
IONIZATION

45. What happens when the orbital frequency becomes large enough to
excite transitions to unbound states?
Ω = Eb
E
Eb
DB, Bertone, Stout and Tomaselli 2021

46. Fermi’s Golden Rule
The transition rate (per unit energy) is given by Fermi’s Golden Rule:
Ω = Eb
ϕ∗(t)
−Eb
E
η!m(E)eiϕ∗(t)
η!m(E)e−iϕ∗(t)
Eb
d `m = dE ⌘`m(E; t)
2
E Eb ⌥ (m mb)⌦(t)
level mixing: hE; `m|V⇤(t)|nb`bmbi
2

47. Pion(t) =
Mc(t)
µ
X
`,m
(m mb) ⌦(t) ⌘`m E
(m)
⇤ ; t
2
⇥ E
(m)
⇤
Ionization Power
Ω = Eb
ϕ∗(t)
−Eb
E
η!m(E)eiϕ∗(t)
η!m(E)e−iϕ∗(t)
Eb
Summing over all bound states gives the total ionization power:
E
(m)
⇤ ⌘ Eb ± (m mb) ⌦(t)

48. Ionization Power
• The orbital energy lost due to ionization can be very large.
| (R )|2
300 200 100 0
0
100
200
R /M
P
ion
/P
GW
Evaluating this numerically, we find
• It has sharp features are specific separations:
R
(g)
⇤ =
M
↵2

4g2
✓
1 +
M⇤
M
◆
n4
b
1/3
, g = 1, 2, · · ·

49. 0 200 400 600 800
10
200
400
+
t [yrs]
R
∗
/M
Backreaction on the Orbit
Ionization can significantly shorten the merger time:

50. Frequency Evolution
We get sharp features in the frequency evolution of the GW signals:
−20 −15 −10 −5 0
t − tmerg
(f
/
)
−8/3
f(2)
f(3)
f(4)
These discontinuities are distinctive signatures of the boson clouds.

51. CONCLUSIONS
AND OUTLOOK

52. • Precise waveform modeling is needed to apply this to GW searches.
• We studied the dynamics of boson clouds in black hole binaries.
• The backreaction on the orbit produces distinct GW signatures.

53. Thank you for your attention!