Black hole Lasers powered by Axion SuperradianT instabilities

BLASTs João G. Rosa

Aveiro University

with Tom Kephart (Vanderbilt University) Phys. Rev. Lett. 120, 231102 (2018) (Editors’ Suggestion)

[arXiv:1709.06581 [gr-qc]]

Café com Física, Universidade de Coimbra, 28 November 2018

Gr@v

Lasers and stimulated emission

Kerr black hole having a BLAST:

superradiant instability

stimulated axion decay

Outline

1.  Black hole superradiance 2.  Axions 3.  Lasing in superradiant axion clouds 4.  Primordial black hole lasers 5.  Fast Radio Bursts 6.  Summary & Future prospects

Black hole superradiance [Zeldovich (1966)]

ω < mΩ

R > 1

Ω

Low frequency waves can be amplified by scattering off a Kerr black hole:

ds2 = −1− 2Mr

Σ

dt2 +

Σ

∆dr2 + Σdθ2

+(r2 + a2)2 − a2 sin2 θ∆

Σsin2 θdφ2 − 4Mra sin2 θ

Σdtdφ

Kerr metric

Event horizon and Cauchy (inner) horizon:

(G = c = = 1)

J = aM ∆ = r2 + a2 − 2Mr Σ = r2 + a2 cos2 θ

r± = M ±

M2 − a2 0 ≤ a ≤ M

Black hole superradiance

Klein-Gordon equation in Kerr space-time:

Separation of variables:

gµν∇µ∇νΦ = 0

stationary and axisymmetric

Φ(t, r, θ,φ) = e−iωteimφS(θ)R(r)

spheroidal harmonics

Superradiance for scalar waves

Schrodinger-like equation

where

d2ψ

dr2∗+

ω2 − V (ω)

ψ = 0

ψ =

r2 + a2R , dr∗ =(r2 + a2)

∆dr

ergoregion

event horizon

Superradiance for scalar waves

I II

ψI = e−i(ω−mΩ)x , ψII = Ae−iωx +Beiωx

General solutions:

Boundary conditions:

Reflection coefficient:

ψI(0) = ψII(0) , ψI(0)− ψ

II(0) = αψI(0)

V (ω) = αδ(x)

+ mΩ(2ω −mΩ)(1−Θ(x))

R =

B

A

2

=α2 +m2Ω2

α2 + (2ω −mΩ)2> 1

Toy model for superradiance

ω < mΩ

I II V (ω) = αδ(x)

+ mΩ(2ω −mΩ)(1−Θ(x))

Toy model for superradiance

In the superradiant regime: •  negative phase velocity: •  positive group velocity:

Waves carry negative energy into the BH Energy and spin extraction from BH

kI = ω −mΩ < 0

vg = dω/dkI = 1

•  Superradiant amplification for different waves:

–  Scalar: 0.3%

– Electromagnetic: 4.4%

– Gravitational: 138%

•  No superradiance for fermions

[Press & Teukolsky (1974)]

Superradiant amplification

[Press & Teukolsky (1972); Cardoso, Dias & Lemos (2004)]

•  surround black hole with mirror

•  multiple superradiant scatterings

•  exponential amplification of signal

•  extract large amount of energy and spin

•  radiation pressure eventually destroys mirror

Black hole bombs

Massive fields can become bound to the black hole: “gravitational atoms”

[Arvanitaki et al. (2009)]

gravita6onal poten6al

Massive black hole bombs

I II

∞

ψI = e−i(ω−mΩ)x , ψII = A sin(ωx)

Bound states satisfy:

In the limit :

ω cot(ωL) + α = i(ω −mΩ)

L

ψI = e−i(ω−mΩ)x , ψII = A sin(ωx)

ωL 1 , α ω

ωR nπ

2L

ω = ωR + iΓ

Γ −ωR(ωR −mΩ)

α

Superradiant instability

Toy model for superradiant instabilities

Φ ∝ e−iωRteΓt

•  Solve radial equation in both regions •  Match asymptotic behaviour in overlap region

•  Possible for small mass coupling:

r − r+ r+

Near region

Far region

Matching asymptotics method

r − r+ l/ω

αµ = GMµ/c 1

Gravitational Hydrogen-like spectrum

Note: matching not possible for extremal BH [JGR (2009)]

superradiant instability

[Detweiler (1980); Furuhashi & Nambu (2004)]

Gravitational “atoms”

ωn µc21−

α2µ

2n2

αµ ≡ GµMBH

c

Γ = −Cln(ωR −mΩ)α4l+5µ

r+ − r−r+ + r−

2l+1

(n, l,m)

αµ 1

Superradiant growth rate

n = 2, l = m = 1

[see review by Brito, Cardoso & Pani (2015)]

[Dolan (2007)]

Non-relativistic scalar clouds n = 2, l = m = 1

v2 (αµ/2)c

Γsa

24α9µ

c3

GM

4× 10−4a

µ

10−5 eV

αµ

0.03

8s−1 ,

r+ 0.1 αµ

0.03

µ

10−5 eV

−1 1 +

1− a2

cm

r0 =

µcαµ 66

αµ

0.03

−1 µ

10−5 eV

−1cmBohr radius:

Event horizon:

Superradiant growth rate:

Superradiant condition implies (for l=m=1):

For stellar mass black holes:

Can use superradiance to probe the low-energy BSM particle spectrum

Astrophysical black hole bombs

ω < Ω ⇒ αµ < 1/2

αµ 0.75

M

M⊙

µ

10−10 eV

CP-violation in Quantum Chromodynamics (QCD):

Neutron electric dipole moment:

Introduce dynamical axion field [Peccei & Quinn (1977)]:

L = −1

4FµνF

µν − θ

32π2Fµν F

µν

θ 10−10

Axions

L = −1

4FµνF

µν − 1

32π2

φ

FφFµν F

µν − 1

2∂µφ∂

µφ− V (φ)

Axion is a pseudo-NG boson of the U(1) Peccei-Quinn symmetry that gets potential from QCD instantons:

Axion can be very light: a

Axions

µ 10−5

Fφ

6× 1011 GeV

−1

eV

φ

V (φ)

[Weinberg; Wilczek (1978)]

The electromagnetic triangle anomaly leads to an axion-photon-photon interaction:

This leads to axion decay into photon pairs:

Axions can account for dark matter!

Axion-photon coupling

Lφγγ =αK

8πFφφFµν F

µν

τφ 3× 1032K−2 µ

10−5 eV

−5Gyr

Misalignment production:

-  axion field initially displaced from origin -  oscillates about origin after QCD transition -  stable cold dark matter condensate

Fine-tuned initial conditions and other production mechanisms yield axion DM range:

Axion dark matter

10−12 eV µ 10−2 eV

Ωφh2 0.11

10−5 eV

µ

1.19

F (Fφ,Θi)Θ2i

[Preskill, Wise & Wilczek; Abbott & Sikivie; Dine & Fischler (1983)]

Axion decay into photon pairs can be stimulated by the passage of a photon:

In dense axion clusters, this can lead to lasing:

-  cross section:

-  photon mean free path:

-  lasing condition:

Lasing axion clusters

Γstφ = 2fγΓφ

σst = Γstφ /Φγ

λγ = 1/(nφσst)

λγ

2R=

µ4τφβR2

24πM 1

[Kephart & Weiler (1987,1995), Tkachev (1987)]

Lasing can occur only for sufficiently dense clusters:

which translates into a large axion degeneracy:

Can lasing occur in superradiant axion clouds?

Lasing axion clusters

ρφρ⊙

6

K4

µ

1 eV

−2

nφ

µ3 2× 1019

K4

µ

1 eV

−6

Boltzmann equation for axion decay/inverse decay:

where:

Lasing equations

dnλ(k)

dt=

dXLIPS

fφ(p)(1 + fλ(k))(1 + fλ(k

))

− fλ(k)fλ(k)(1 + fφ(p))

|M|2

ni =

d3ki(2π)3

fi(ki)

Simplified cloud model:

-  “2p”-toroidal axion cloud (flat space, non-relativistic) -  homogeneous and isotropic phase space distributions

Lasing equations

pφ αµ

2µc , pγ µc

2, ∆pγ αµ

2µc

dnγ

dt= Γφ

2nφ

1 +

8π2

µ3βnγ

− 16π2

3µ3

β +

3

2

n2γ

spontaneous decay

stimulated decay

photon annihilation

“active” axions

“sterile” axions

We also need to include:

•  axion source from BH superradiant instability •  photon escape rate:

Photon-axion dynamics:

Lasing equations

Γe = c/(√5r0)

dNφ

dt= ΓsNφ − Γφ

Nφ(1 +ANγ)−B1N

2γ

,

dNγ

dt= −ΓeNγ + 2Γφ

Nφ(1 +ANγ)−BN2

γ

,

A = 8α2µ/25 , B1 = 2α4

µ/75 , B2 = 2α3µ/25 , B = B1 +B2

Γφ Γs Γe

Numerical solution

µ = 10−5 eV

K = 1

MBH = 8× 1023 kg

a = 0.7

αµ 0.03

N cφ =

Γe

2AΓφ

N cγ =

Γs

AΓφ

Laser pulses Neglecting annihilation, close to lasing threshold:

where:

X = Nφ/Ncφ − 1 1 , Y = Nγ/N

cφ 1

d2 log Y

du2= η − 1

2Y

u = Γet , η = Γs/Γe 1

Y =

Y0eηu

2/2 , Y 2η

C cosh−2√

C(u− umax)/2

, Y 2η

BLAST properties For the first laser pulses:

LB 2× 1042

K2a

10−5 eV

µ

2 αµ

0.03

7

ξ

100

erg/s

τB 1√a

10−5 eV

µ

αµ

0.03

−9/2

ξ

100

−1/2

ms

νB 1.2 µ

10−5 eV

GHz

ξ = log (Γs/Γφ)

107− 4 logµ/10−5 eV

+ 8 log (αµ/0.03) + log

a/K2

Plasma effects BLASTs typically exceed BH Eddington luminosity

i.e. radiation pressure blows away surrounding plasma.

However, electric field within cloud can exceed the Schwinger threshold for e+e- production:

LEdd 1038 (M/M⊙) erg/s

|E| ∼ Ec

µ

10−5 eV

αµ

0.03

L

1043 erg/s

1/2

Plasma effects Rate of Schwinger electron-positron plasma production:

Expect plasma to reach critical density in a single pulse:

and effective photon plasma mass blocks axion decay.

BLASTs are short electromagnetic bursts!

dne

dt=

α0π2 |E|2e−πEc/|E| 1056

|E|2

E2c

e−πEc/|E| m3s−1

νp =

nee2

4π20me 1

ne

1016 m−3

1/2GHz

Plasma effects Electrons and positrons eventually annihilate:

allowing lasing to restart.

Expect BLASTs to repeat!

τann 4 ne

1016 m−3

−1hours

Constraints 1. Critical cloud mass/spin for lasing:

BLASTs only possible for

2. Non-linear axion self-interactions quench superradiance (“bosenova”):

Jc

φ

JBH

0.06

aα3µK2

µ

10−8 eV

−2 1

µ 10−8 eV MBH 10−2M⊙Primordial Black Holes

φc Fφ αµ 0.03K

[Kodama & Yoshino (2012-2015)]

Primordial black holes Large density fluctuations collapse directly into BHs once they re-enter the horizon in radiation era:

Possible mechanisms: (multi-field) inflation, curvaton, phase transitions

PBH mass comparable to mass within Hubble horizon:

δρ/ρ 0.1

M =c3t

G 1025g−1/2

∗

T

300 GeV

−2

kg

Primordial black holes PBHs heavier than 1012 kg survive until today and may contribute to dark matter!

[Carr, Kuhnel & Sandstad (2017)]

Primordial black holes PBHs are born with no spin and accrete mostly in spherical configuration [Ali-Haimoud & Kamionkowski (2017)]

However, PBHs can merge into spinning PBHS!

e.g. two equal Schwarzschild BHs merge into BH with spin [Scheel et al. (2009)]

PBH merger rate in clustered scenario: [Clesse & Garcia-Bellido (2016)]

Clusters, dwarf spheroidal galaxies:

af 0.7

Γtotalcapt 3× 10−9fDMδloc yr−1Gpc−3

δloc 109 − 1010

Fast Radio Bursts If axions account for most of the dark matter:

They can produce repeating BLASTs for PBHs with mass

BLAST properties:

longer BLASTs are fainter and yield short-lived continuous lasers (1-100 years)

shorter BLASTs are brighter and yield long-lived repeating fast radio bursts (104 years)

µ ∼ 10−5 eV

MPBH (2− 8)× 1023 kg

νB ∼ 1 GHz LB ∼ 1039 − 1042 erg/s τB ∼ 100− 1 ms

Fast Radio Bursts If axions account for most of the dark matter:

They can produce repeating BLASTs for PBHs with mass

BLAST properties:

longer BLASTs are fainter and yield short-lived continuous lasers (1-100 years)

shorter BLASTs are brighter and yield long-lived repeating fast radio bursts (104 years)

µ ∼ 10−5 eV

MPBH (2− 8)× 1023 kg

νB ∼ 1 GHz LB ∼ 1039 − 1042 erg/s τB ∼ 100− 1 ms

Fast Radio Bursts More than 30 FRBs with these properties have been observed in the past decade [Lorimer et al. (2007)]

FRB 121102 repeats and is localized within a faint dwarf galaxy [Chatterjee et al.; Marcote et al.; Tendulkar et al. (2017)]

Possible mechanisms: compact object mergers; NS collapse; energetic pulsars/magnetars…

Could BLASTs account for some of these FRBs?

QCD axion exists and is the dark matter

Sub-terrestrial PBHs exist and merge

Predictions •  Up to 105 active FRBs across the sky (PBHs <10% of DM)

•  X-rays afterglows from e+e- annihilation

•  Gravitational waves from mergers

•  Gravitational waves from bosenova between bursts

•  DM axion-photon conversion in galactic B-field @ SKA [Kelley & Quinn (2017)]

•  QCD axion with mass ~10-5 eV and gφγγ∼10-15 GeV-1 [ADMX, X3, CULTASK, MADMAX, ORPHEUS]

Summary

•  Superradiant instabilities can produce dense axion clouds around spinning black holes

•  Stimulated axion decay leads to extremely bright lasing bursts (BLASTs)

•  Tantalizing agreement with observed Fast Radio Bursts for cosmological axions and primordial black holes

•  Lasing may occur for other light BSM bosons?

Thank you!