The Fate of Axion Stars
Hong Zhang The Ohio State University In collabora*on with Eric Braaten and Abhishek Mohapatra PRL 117, 121801 (2016)! PRD 94, 076004 (2016)! arXiv:1609.05182!
Outline ² Axions ² (Dilute) Axion Star ² Dense Axion Star
PRL 117, 121801 (2016)!
² Observables
arXiv:1609.05182!
² Axion EFT
PRD 94, 076004 (2016)!
² Summary 1
Axions • Peccei-Quinn U(1) symmetry solves strong CP problem! Peccei & Quinn (1977)
• Introduces a Goldstone boson -- Axion! Weinberg (1978), Wilczek (1978)
• Strongly motivated candidate for cold dark matter.! Lect. Notes Phys. 741 (2008) A recent review: Kim & Carosi (2010) 2
Relativistic Axions Real pseudoscalar field! µ 1 L = 2 @µ @
Energy scale below 1GeV!
V( )
Two models for potential ! • Instanton!
V( ) = m2a fa2 [1
cos( /fa )]
ma : axion mass! fa : axion decay constant! • Chiral!
V( ) =
m2⇡ f⇡2
1
1
z = mu /md ⇡ 0.48
4z 2 sin ( /2fa ) 2 (1 + z)
1/2
!
3
Relativistic Axion Potential Periodic potentials! 4
Ratio ~ 1.5!
Chiral!
3
V /ma fa
2 2
V m2a fa2
V( ) = V( + 2⇡fa )
Instanton! 2
1
0
-6
-4
-2
0
φ /fa
/fa
2
4
6 4
Parameters & Current Constraints • Two parameters in relativistic axion Lagrangian: !
ma and fa • Not independent, related by QCD!
m2a fa2
z 2 2 = m ⇡ f⇡ 2 (1 + z)
ma fa = (80 MeV )
2
z = mu /md ⇡ 0.48
• Constraints from astrophysics & cosmology ! 6 8 13
10 GeV < fa < 10 GeV
10
In this talk, I choose !ma
= 10
2
eV
Tiny Mass !!
Very weak self-interaction ! 4
eV < ma < 10
eV
5
Loop Contribution is Small Each loop is suppressed by !
(ma /fa )2 ⇠ 10
48
• Diagrams with loops can be safely ignored.!
6
Axion-Photon Coupling • Very weak coupling!
cem ↵ µ⌫↵ Lem = ✏ Fµ⌫ F↵ 16⇡fa cem ⇠ 1 Model dependent! Suppressed by ! fa
.
⇠ 1011 GeV
• Decay rate into two photons! a
=
c2em ↵2 m3a . 3 2 256⇡ fa
Photon energy:!
⇠ 1036 years 10 Age of Universe! ⇠ 10 years
Axion lifetime!
ma /2 ⇠ 10 GHz Radio frequency 7
Axion Cosmology • Cold dark matter axions are produced abundantly
at QCD phase transition scale T ~ 1 GeV ! Non-relativistic axion production mechanism! For more details, see Lect. Notes Phys. 741 (2008)
Ø Vacuum misalignment!
Coherent
Preskill, Wise & Wilczek (1983) Abbot & Sikivie (1983) Dine & Fischler (1983)
Ø Cosmic string decay!
Incoherent
Davis (1986) Hararie & Sikivie (1987)
3
Occupation number! na dB |T =1GeV
⇡ 1058
Sikivie & Yang PRL (2009)
Form Bose-Einstein condensate if can be effectively thermalized! 8
Gravitational Thermalization • Axion self-interaction may be too weak to thermalize axions! • Gravitational interaction can thermalize axions! Sikivie & Yang PRL (2009)
Ø Bring initially incoherent axions into coherence! Ø Keep the axion field as a Bose-Einstein Condensate
as the Universe evolves! • Correlation length! Galactic scale?!
Sikivie & Yang PRL (2009)
Stellar scale due to attractive self-interaction?! Guth, Hertzberg & Prescod-Weinstein PRD (2015)
Is there a (meta)stable axion star solution?
9
Outline ² Axions ² (Dilute) Axion Star ² Dense Axion Star
PRL 117, 121801 (2016)!
² Observables
arXiv:1609.05182!
² Axion EFT
PRD 94, 076004 (2016)!
² Summary 10
Non-relativistic EFT (Part I) L=
1 2 @µ
@
µ
Real Scalar!
V( )
Chavanis PRD (2011) Chavanis & Delfini PRD (2011)
Instanton potential / Chiral potential!
Naïve non-relativistic reduction! Complex scalar!
1 ⇥ (r, t) = p (r, t)e 2ma
ima t
+
⇤
(r, t)e
+ima t
Ignore all terms with rapid oscillating phase !
Le↵ =
1 2i
Ve↵ = ma
⇣
⇤ ⇤
˙
˙⇤
⌘
1 r 2ma
⇤
·r
⇤
Ve↵
1 ( ⇤ )2 1 ( ⇤ )3 + + ... 2 4 16 fa 288 ma fa Expand by !
⇤
ma fa2
11
Non-relativistic EFT (Part I) L=
1 2 @µ
@
µ
Real Scalar!
V( )
Chavanis PRD (2011) Chavanis & Delfini PRD (2011)
Instanton potential / Chiral potential!
Naïve non-relativistic reduction! Complex scalar!
1 ⇥ (r, t) = p (r, t)e 2ma
ima t
+
⇤
(r, t)e
+ima t
Ignore all terms with rapid oscillating phase !
Le↵ =
1 2i
Ve↵ = ma
⇣
⇤ ⇤
˙
˙⇤
⌘
1 r 2ma
⇤
·r
Ve↵
1 ( ⇤ )2 1 ( ⇤ )3 + + ... 2 4 16 fa 288 ma fa
Attractive interaction!!
Expand by !
⇤
Dilute
limit!
⇤
ma fa2
12
Dilute Axion Stars Assume:! • Instanton potential, dilute axion limit! • Newtonian gravity! • Spherically symmetric! ⇥ 0 ⇤ r2 ⇤ KineZc + (Ve↵ ( ) + ma = (µ ma ) , pressure 2ma
r2 = 4⇡Gma
⇤
.
Self-interacZon
1.0
example
0.8
| (r)|2 | (0)|2
Gravity
0.6
Dilute!
0.4
0.2
0.0 1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
normalized radius
13
(First) Critical Point • Heavier dilute axion stars have smaller radii.! • Critical mass: beyond which the kinetic pressure
cannot balance the attractive self-interaction and gravity!
stable!
critical pt!
unstable!
Critical point:!
p M⇤ = 10.2fa / Gm2a
= 6 ⇥ 10
R⇤ = 3 ⇥ 10
14 4
M
R
= 200 km 14
Formation of Dilute Axion Stars • Dilute axion stars can be produced in early universe.! • Vacuum misalignment mechanism produces coherent and
non-relativistic axions.! • Spatial fluctuations in the axion field evolve into gravitationally
bound “miniclusters” of axions.! • Gravitational thermalization drives the axion minicluster to
form a dilute axion star.! • Dilute axion stars attract more axions and gradually reaches
the critical mass. !
15
End of Dilute Axion Stars • Dilute axion stars collapse when its mass exceeds
the critical mass,
• !What is the remnant?!
Less massive
dilute axion star?! by emitting
extra axions!
Black hole: Schwarzchild radius is ~20 orders of magnitude smaller!
Dense axion star?! Radius is 5 orders
smaller!
Chavanis arXiv: 1604.05904 Helfer et al. arXiv: 1609.04724
Eby et al, arXiv: 1608.06911 16
Outline ² Axions ² (Dilute) Axion Star ² Dense Axion Star
PRL 117, 121801 (2016)!
² Observables
arXiv:1609.05182!
² Axion EFT
PRD 94, 076004 (2016)!
² Summary 17
Non-relativistic EFT (Part II) Le↵ =
1 2i
⇣
⇤
˙
⌘
˙⇤
• Dilute axion field!
Ve↵ = ma
⇤
1 r 2ma
⇤
·r
Ve↵
1 ( ⇤ )2 1 ( ⇤ )3 + + ... 2 4 16 fa 288 ma fa
• In dense axion field (
⇤
Dilute
limit!
) ⇠ ma fa2 , must keep all orders !
Both instanton and chiral potential can be summed to all orders! Instanton potential:!
Ve↵ (
⇤
)=
⇤ 1 m 2 a
+ m2a fa2
⇥
1
J0 (2
⇤
/ma fa2 )
Eby, Suranyi, Vaz & Wijewardhana (2015)
18
⇤
Non-relativistic Instanton Potential ⇤
Ve↵ (
)=
⇤ 1 m 2 a
+ m2a fa2
[Vclass -
∗ 2 2 1 __ m aψ ψ] /m a f a 2
2.5
⇥
1
J0 (2
⇤
/ma fa2 )
Not periodic! Decreasing Amplitude!
2
1.5
1
0.5
0 0
Instanton 3
6
9 12 2 1/2 (2ψ ψ/ma fa )
15
⇤
18
∗
Braaten, Mohapatra, HZ, PRD (2016)!
19
Dense Axion Stars Assume:! • Instanton potential! • Newtonian gravity! • Spherically symmetric! Compare axion number density! • Dilute axion star: Gaussian-like! • Dense axion star: almost flat, with a fast-dropping edge! 1.0
example 0.8
0.6
| (r)|2 | (0)|2
0.4
0.2
0.0
example
1.0
0.8
0.6
0.4
Dilute! 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1
Normalized normalizedradius! radius
0.2
0.0 0.0
= 6.58 0
Dense! 0.2
0.4
0.6
0.8
Braaten, Mohapatra, HZ, PRL (2016)!
1.0
20
Self-interaction Force r2 2ma
+
⇥
0 (Ve↵ ( ⇤
) + ma
⇤
ma ) , r2
= (µ
• Self-interaction force
F = int (mean-field pressure)!
00 Ve↵ ( ⇤
d ) ( dr
= 4⇡Gma ⇤
⇤
)
0.01
Smaller than 0!
Outwards! 0.005
More branches!
∗
’’ 2 Veff (ψ ψ) f a
Fint 0
First dense axion
star branch!
-0.005
Inwards! -0.01 0
10 15 ∗ 2 1/2 Normalized f ) (2ψ ψ/mdensity! a a
5
20
25
Braaten, Mohapatra, HZ, PRL (2016)!
21
.
Forces Balancing r2 2ma
+
⇥
0 (Ve↵ ( ⇤
) + ma
⇤
= (µ
ma ) , r2
= 4⇡Gma
• Recall in dilute axion star, kinetic pressure balances gravity
and self-interaction force! KineZc pressure
Self- interacZon Gravity KineZc pressure
Gravity
Dense!
Self- interacZon
• In dense axion star!
Bulk: ! Ø self-interaction force balances
gravity, ! Ø kinetic pressure ~ 0! Ø wave-function is almost flat! Surface:! Ø large kinetic pressure needed
to balance the other two,! Ø wave-function drop rapidly! Braaten, Mohapatra, HZ, PRL (2016)!
22
⇤
.
0.01
Forces Balancing 0 + (Ve↵ (
⇤
) + ma
⇤
∗
⇥
Outwards!
0.005 ’’ 2 Veff (ψ ψ) f a
r2 2ma
Fint
= (µ
0
ma ) , r2
= 4⇡Gma Inwards!
• In dense axion star! 2 1/2
∗
a a
Self- interacZon Gravity KineZc pressure
Gravity
Dense!
Self- interacZon
.
-0.005
• Recall in dilute axion star, kinetic pressure balances gravity
-0.01 0 10 20 5 15 and self-interaction force! (2ψ ψ/m f ) KineZc pressure
⇤
Bulk: ! Ø self-interaction force balances
gravity, ! Ø kinetic pressure ~ 0! Ø wave-function is almost flat! Surface:! Ø large kinetic pressure needed
to balance the other two,! Ø wave-function drops rapidly! Braaten, Mohapatra, HZ, PRL (2016)!
23
25
Thomas-Fermi Approximation • When the surface thickness is small compare to the bulk,
Thomas-Fermi approximation can be applied.!
r2 2ma
+
⇥
0 (Ve↵ ( ⇤
r2 = 4⇡Gma
⇤
) + ma
.
⇤
= (µ
ma ) ,
Greatly simplifies!
• Interaction force (mean-field pressure) exactly balances
gravitational force! • Not applicable to small dense axion star, in which the surface
thickness is important! Braaten, Mohapatra, HZ, PRL (2016)!
24
Radius vs. Mass Sun!
1
dilute axion star! -2
10
Moon! Halley’s! comet!
10-4
R/R
Earth!
critical pt!
10-6
unstable! 10-8
! ? e l stab
10-10
critical pt! 10-12 10-21
10-18
10-15
10-12
10-9
10-6
10-3
1
M/M
• 2nd critical point! • Newtonian gravity is justified! • Heavier dense axion stars have larger radii! Braaten, Mohapatra, HZ, PRL (2016)!
25
Formation of Dense Axion Stars • Possible remnants of dilute axion stars collapsing! 1
dilute axion star! 10-2
critical pt!
R/R
10-4
Bosenova?!
10-6 10-8 10-10 10-12 10-21
10-18
10-15
10-12
10-9
10-6
10-3
1
M/M
Braaten, Mohapatra, HZ, PRL (2016)!
26
Outline ² Axions ² (Dilute) Axion Star ² Dense Axion Star
PRL 117, 121801 (2016)!
² Observables
arXiv:1609.05182!
² Axion EFT
PRD 94, 076004 (2016)!
² Summary 27
Axion Detection • Depends on the tiny axion-photon coupling! cem ↵ µ⌫↵ Lem = ✏ Fµ⌫ F↵ . 16⇡fa
cem ⇠ 1
Model dependent!
Suppressed by ! fa
a
=
c2em ↵2 m3a . 3 2 256⇡ fa
11
⇠ 10 GeV
⇠ 1036 years 10 Age of Universe! ⇠ 10 years
Axion lifetime!
• Direct detection, indirect detection and laser experiment.! Lect. Notes Phys. 741 (2008) A recent review: Kim & Carosi (2010)
28
Indirect Detection • Detect radio-frequency photons! • Currently no experiment available!
(tiny
a
coupling)!
• Two photon-production channels! Ø One-step: NR axions !! Ø Two-step: NR axions
!
relativistic axions ! !
29
1st Channel: NR Axions to Photons
ma ⌫0 = 2
ma ⌫=5 2
ma ⌫=3 2
...
• Odd-integer harmonics of the fundamental radio frequency.! After broadening and red-shifting!
1 3 5
⌫/⌫0
Unique feature of axions !!
Braaten, Mohapatra, HZ, arXiv:1609.05182!
30
2nd Channel • Two relativistic axions production! 2 2 suppressed by one power of !(ma /fa ) ⇠ 10
4a ! 2a
...
6a ! 2a
...
• More relativistic axions!
48
suppressed by more powers!
• Much weaker photon signal than 1st channel! Braaten, Mohapatra, HZ, arXiv:1609.05182!
31
Two Types of Sources • Continuous photon emission! Ø Stable axion stars! • Catastrophic phenomenon: ! a lot of energy released in a short time! Ø Collapse of dilute Axion stars! Ø Collision of an axion star with a neutron star! Ø …!
32
Emission From Dense Axion Star 10
6 6
10
Relative to!
a!
Axion to photon
decay rate!
4a→2a
6a→2a
10
3
3
| dN/dt | / NΓa
10
1
10 10
a!2
60
eV
a→γγ
0
10
• Solid:
Instanton potential!
3a→γγ 5a→γγ 3a→γγ
3 -3
• Dashed:
Chiral potential
TF approximation!
10
6
5a→γγ
-6
10 -21 10
= 10
-18
10
-15
10
-12
10
-9
10
M/MM/M
-6
10
4a ! 2a, 6a ! 2a, 8a ! 2a
10
-3
10
0
are all zero for instanton potential!
Braaten, Mohapatra, HZ, arXiv:1609.05182!
33
Emission From Dilute Axion Star Relative to!
110
a!2
10
10
-50 50
60
eV
• Solid:
Instanton potential!
4a→2a 5a→γγ
-75 75
• Dashed:
Chiral potential!
6a→2a
10
= 10
3a→γγ
10
10
Axion to photon
decay rate!
a→γγ
-25 1025
| dN/dt | / NΓa
10
0
a!
-100 100
10
-14
10
M/M
-13
10
M/M
4a ! 2a, 6a ! 2a, 8a ! 2a
are all zero for instanton potential!
Braaten, Mohapatra, HZ, arXiv:1609.05182!
34
Photon Flux Estimate • Single axion decay to two photons is independent of the
configuration. !
• Solar system 3
DM density ~!0.3GeV/cm 1068 axions! Total DM in Solar system! ⇠ 0.01M radius ~ 125,000 AU,
22 Photon production rate! ⇠ 10 /sec
Energy released: ! ⇠ 109 GeV/sec ⇠ 0.4 Watt
• Milky way:! Total
DM:!
⇠ 1012 M
1082 axions!
Largest 1st branch dense axion star has ~1070 axions!
⇠ 1011 GeV/sec ⇠ 40 Watt
35
Axion Stars are Not So Bright ! PRL 117, 121801 (2016)
Editors’ suggestion!
Picture chosen by PRL! 36
Hydrogen Axion Star ? • Hydrogen gas is captured by the gravitational potential well
of axion star, forming dense metallic fluid state.! • Electron interacts with axion, generating heat, resulting in
blackbody radiation with peak in the UV region.! • Energy released: !
1013
⇣ m ⌘4 a W⇥ 5 meV
• The signal should be readily visible to current high-resolution
telescopes.! Bai, Barger and Berger, JHEP (2016)!
37
Catastrophic Phenomena Fast Radio Burst (FRB)! • A ultra-fast (milli-sec) burst of photons in radio frequency.! • Nothing similar observed in optical, X rays and ϒ rays! • Since 2007, 17 events have been reported.! • Estimated rate ~ 104 sky -1 day -1 ! • Reported frequency is 1.4 GHz (telescope design)! • Extra-galactic sources from dispersion measure! • Energy released up to 1040 erg ~ 10 -14 M¤ (If isotropic)! • Strong linear polarization observed.! Recent review: Katz, arXiv:1604.01799 Online database: hbp://www.astronomy.swin.edu.au/pulsar/frbcat
38
Fast Radio Burst
Figure from Nature 530, 453 (2016)
39
Are Axion Stars an Explanation? ü Observed frequency: 1.4 GHz ! 10 6 eV < ma < 10 2 eV 0.2 GHz < ⌫ < 2400 GHz Also explains why such burst is not observed in other bands.! ü Total energy released: up to!⇠ 10 14 M • Dilute axion star critical mass ! 6 ⇥ 10 • Dense axion star mass!
14
M 10 20 M to 2 M
ü Time duration: ~ 1 ms! • Dilute axion star critical radius: 200 km! • Dense axion star radius: 1m to 10 km! ü Polarized photons! Axions in axion stars are in coherence!
40
Scenarios with Axion Stars • Collision of a dilute axion star with a neutron star! Coherent electric dipole radiation! Ø From electrons in atmosphere!
Iwazaki, hep-ph/9908468!
Ø From neutrons in outer core!
Raby, PRD (2016)!
• Collapse of dilute axion stars above the critical mass! Tkachev, JETP Lett. (2014)!
• Collision of two axion stars!
Eby et.al., arXiv:1701.01476!
• Collision of a dense axion star with a neutron star! 41
Observe Odd-integer Harmonics • One unique feature of axion stars:
odd-integer harmonics of the fundamental radio frequency.! • Can we observe the fast radio burst at other frequencies? ! 1.4 GHz,
3 × 1.4 GHz,
5 × 1.4 GHz …!
or! 1/3 × 1.4 GHz,
1.4 GHz,
5/3 × 1.4 GHz …!
Many possible combinations! • Need more events in more frequency windows.!
42
Outline ² Axions ² (Dilute) Axion Star ² Dense Axion Star
PRL 117, 121801 (2016)!
² Observables
arXiv:1609.05182!
² Axion EFT
PRD 94, 076004 (2016)!
² Summary 43
Axion EFT • Relativistic Axions: real scalar!
L=
1 2 @µ
@
µ
V( )
Instanton or chiral
potential!
• Non-relativistic axions: complex scalar!
He↵
1 = r 2ma
⇤
· r + Ve↵ (
⇤
)
Ø Integrate out axion mass scale! Ø Much simpler, but equally accurate in the NR limit! • Need to find the NR potential !Ve↵ (
⇤
) 44
Naïve NR Reduction • Zeroth order approximation!
⇤ 1 ⇥ ima t ⇤ +ima t (r, t) = p (r, t)e + (r, t)e 2ma ⇤ 2 ⇤ 3 1 ( ) 1 ( ) Ve↵ = ma ⇤ + + ... Dilute! 2 4 16 fa 288 ma fa ⇥ ⇤ ⇤ ⇤ 2 2 ⇤ 2 1 ) = 2 ma + ma fa 1 J0 (2 /ma fa ) Dense! Ve↵ ( • Used to get dilute and dense axion star solutions! • Not considering virtual axions!
(3ma , p) Braaten, Mohapatra, HZ, PRD (2016)!
45
Match the amplitude • Matching low-energy scattering amplitudes.! • Includes all virtual axion contributions! 2 (m /f ) • Only tree diagram: loops are suppressed by ! a a ⇠ 10
48
• Example: 3 to 3 scattering! Braaten, Mohapatra, HZ,
PRD (2016)!
RelaZvisZc coupling
Relativistic
field theory!
=! Axion EFT! Match and obtain NR coupling
46
Match Low-power Couplings • Expand the NR potential!
Ve↵ (
⇤
• Check
) = ma
⇤
+
(v2 , v3 , v4 , v5 )
NR reduction:!
( -1,
With matching:! ( 1, Deviation:!
m2a fa2
1 X
vn 2 (n!) n=2
✓
⇤
2ma fa2
◆n
for instanton potential 1,
-1,
1)
! !
1.125, 2.25, 1.76)
( 0, -189%, -56%, -43%)
!
Contribution of virtual axions is important !! Braaten, Mohapatra, HZ, PRD (2016)!
.
47
Dense Regime • Cannot truncate the power expansion! • Impossible to extract all couplings by matching (infinitely many) ! • One scheme: include more and more virtual axion propagators
in the matching! Naïve NR reduction!
p
Match diagrams with no virtual propagator!
n
n n−p
1st improvement! Match diagrams with 0 or 1 virtual propagator! p
n
q
n n−p
n−q
Braaten, Mohapatra, HZ, PRD (2016)!
48
Summary • Gravity can thermalize axions toward Bose-Einstein
condensates and form dilute axion stars.! • A dilute axion star accumulates axions and collapses once its
mass exceeds the critical mass!10 14 M • Dense axion star is a possible remnant.! • Catastrophic phenomena involving axion stars can release
a large amount of coherent radio-frequency photons in a
very short time, which may explain fast radio burst.! • The photons in odd-integer harmonics of a fundamental
radio frequency are a unique signature of axions.! 49
