Axion stars: formation, death and astrophysicalimplications
I. Tkachev
INR RAS, Moscow

20 May 2019, ICRR, Tokyo

I. Tkachev

Axion stars

1 / 30Outline

Formation of Bose Stars
D.Levkov, A.Panin, & IT, PRL 121 (2018) 151301

Axion Bose Star collapse
D.Levkov, A.Panin, & IT, PRL 118 (2017) 011301

Cosmological and astrophysical implications

I. Tkachev

Axion stars

2 / 30Bose-stars

Bose star is self-gravitating field configuration in the lowest energystate.
Ruffini & Bonazzola, Phys. Rev. 187 (1969) 1767
May appear in Dark Matter models with light Bose particles.
Mainstream candidates - QCD axion or ALP in general:
Axion stars
IT, Sov. Astron. Lett. 12 (1986) 305

Vast literature, but little attention to the problem of their formation.

I. Tkachev

Axion stars

3 / 30Bose-star formation.
Interactions are needed to form Bose condensate
But ALP couplings are extremely small
String axions

QCD axions
Solve strong CP problem

Appear in string models

CDM: m ≈ 26 µeVFuzzy DM: m ∼ 10−22 eV

λ ∼ 10−50

λ ∼ 10−100

Relaxation time is enhanced due to large phase space density f
IT, Phys. Lett. B 261 (1991) 289

τR−1 ∼ σ v n f

where f ∼n
1(mv)3

which is still not enough to beat small λ (except in rare axion miniclusters)
I. Tkachev

Axion stars

4 / 30Bose condensation by gravitational interactions
Are we crazy?
No




f  1 — classical fieldsv  1 — nonrelativistic approximationGravity but no other interactions

ψ(t, x)U (t, x)Field equations for light DM (Scrödinger-Poisson system)i∂t ψ = −∆ψ/2m + mU ψ
Bose star is a stationary solution:∆U = 4πG(m|ψ|2 −hρi)ψ = ψs (r)e−iωt| {z }ρSolving these equations in time, we find Bose condensation!

I. Tkachev

Axion stars

5 / 30Bose condensation by gravitational interactions
Are we crazy?
No





f  1 — classical fields

v  1 — nonrelativistic approximationGravity but no other interactions

ψ(t, x)U (t, x)Field equations for light DM (Scrödinger-Poisson system)i∂t ψ = −∆ψ/2m + mU ψBose star is a stationary solution:∆U = 4πG(m|ψ|2 −hρi)ψ = ψs (r)e−iωt| {z }ρSolving these equations in time, we find Bose condensation!

I. Tkachev

Axion stars

5 / 30Bose condensation by gravitational interactions
Are we crazy?
No





f  1 — classical fieldsv  1 — nonrelativistic approximationGravity but no other interactions

ψ(t, x)U (t, x)Field equations for light DM (Scrödinger-Poisson system)i∂t ψ = −∆ψ/2m + mU ψ
∆U = 4πG(m|ψ|2 −hρi)| {z }
ρBose star is a stationary solution:ψ = ψs (r)e−iωt
| |.2

| |.1

s004r

Solving these equations in time, we find Bose condensation!
I. Tkachev

Axion stars

5 / 30Initial conditions
t=0

Maximally mixed (virialized) initial state.2

Subsequent evolution in kinetic regime.02⇒ Random initial field:−p2 /2(mv0 )2

ψp ∝ e|{z}momentum
distribution−hψ(x)ψ(y)i ∝ e(x−y)2l2
coh×xiApe| {z }
randomphaseslcoh =02
mv0and R  (mv0 )−1 is assumed
I. Tkachev.1ylcoh ∼ (mv)−1  R(mv 2 )−1  τgr|ψ|Axion stars

Virialized DM halo
6 / 30Time evolution

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Axion stars

7 / 30Time evolution
Maximum field value over the simulation box
t = 1.3 · 106

maxx |ψ|.3.2.1yτgr

| |

.1.02

005 · 105

106

t̃

x0

This is not a Jeans instability

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Axion stars

8 / 30It’s a Bose start = 1.3 · 106| || |.2| |.1sy.1.02x0004r

We observe formation of a Bose star at t = τgr

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Axion stars

9 / 30Bose star appearance: another signature
Energy distribution at different moments of time
RR 1 ∗dn22
ψ (t, x)ψ(t + t1 , x) eiωt1 −t1 /τ1F (ω, t) ≡= d3 x dt
2πdω

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10 / 30

Kinetics
Landau equation — derivation
Perturbative solution of Schrödinger-Poisson equation
Kinetic approximations (mv)−1  x, (mv 2 )−1  tCompute Wigner distribution
Rfp (t, x) = d3 y e−ipy hψ(x + y/2)ψ ∗ (x − y/2)irandom phase average
e.g. Zakharov, L’vov, Falkovich ’92∂t fp +p
∇x fp − m∇x Ū ∇p fp = St fpm2F (ω)

LandauSP0

Good agreement of lattice and kinetic F (ω)

t ∼ τgr0

0.5
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1Axion stars

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Kinetics
Landau equation — derivation
Perturbative solution of Schrödinger-Poisson equation
Kinetic approximations (mv)−1  x, (mv 2 )−1  tCompute Wigner distribution
Rfp (t, x) = d3 y e−ipy hψ(x + y/2)ψ ∗ (x − y/2)irandom phase average
e.g. Zakharov, L’vov, Falkovich ’92∈pfp∂t fp +
∇x fp − m∇x Ū ∇p fp = St fp ∼mτkin ← relaxation time
fp3 ← Bose amplification√4 2bTime to Bose star formation: τgr = b τkin =σgr v nf
↑O(1) correction
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Time to Bose star formation
τgr =√4 2bσgr vnf

Rutherford cross section: σgr ≈ 8π(mG)2 Λ/v 4

Λ = log(mvR)
Coulomb logarithm

Average phase-space density: f = 6π 2 n/(mv)3τgr =√b 2mv 612π 3 G2 Λn2=√b 2π3G2 m5 Λf −2

Strongly depends on local quantities: n, v, fInvolves global logarithm Λ = log(mvR)

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Time to Bose star formation
Kinetic scaling of τgr with parameters| |y.1.02x106

τgr · mv 2t = 1.3 · 106

Gaussianδ

3101060108

m2 v 8 /G2 n2 Λ22Gaussian: fp ∝ |ψp |2 ∝ e−p /(mv0 ) , b ≈ 0.9
δ: fp ∝ δ(|p − p0 |), b ≈ 0.6
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Bose star formation in halo/minicluster
t = 1250

t = 250|ψ|y32Large box ⇒ Jeans instability⇒ minicluster

1xx0

τgr =2106√b 2mv 6

12π 3 G2 Λn2

τgr · mv

Virial velocity: v 2 ∼ 4πGmnR2 /3

τgr ∼103
106

0.05 RΛv(Rmv)3

τgr  R/v ← free-fall time
Rmv ∼ 1 — condense immediately

108

m2 v 8 /G2 n2 ΛI. Tkachev

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Models: fuzzy dark matter
String axions
6τgr ∼ 10 yrm10−22 eV

3 

6 v
30 km/s

0.1 M /pc3

2

ρFornax dwarf galaxyv ∼ 11 km/s
ρ ∼ 0.1 M /pc3
τgr ∼ 1000 yr
Universe filled with Bose stars!

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Axion stars

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Models: QCD axion

Peccei-Quinn phase transition at T ∼ faAxion potential switches on at QCD epoch

V (a) = fa ma (T ) [1 − cos(a/fa )]V (a) = fa ma (T ) [1 − cos(θ)]θ ≡ a/fa

PQ phase transition before inflation is disfavored
PQ phase transition after inflation:
Three contributions to axion DM
Coherent oscillations of axion field (misalignment angle)
Decay of PQ strings
Decay of QCD axion domain walls
M. Nagasawa and M. Kawasaki, Phys. Rev. D50 (1994) 4821
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QCD axion cosmology
PQ phase transition after inflation → Miniclusters

After phase transition 0 < θ < 2π from horizon to horizon,but θ ≈ const on a horizon scale lH
Peculiar initial amplitude of oscillations when ma turns on
Dark Matter should be very clumpy

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QCD axion cosmology
PQ phase transition after inflation → Miniclusters

Mass scale of the clumps is set by M ∼ 10−11 M , which is DM mass within
horizon at Tosc ≈ 1 GeV
Naively, initial DM density contrast is δρa /ρa ≡ Φ ∼ 1In fact, very dense objects can form, Φ  1, since for θ ∼ 1 the axion attractive
 2
self-coupling is non-negligible,θθ4V (a) = m2 fa2−+ ...
24!
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QCD axion cosmology
Minicluster evolution around equality

Minicluster Formation at QCD

A clump becomes gravitationally bound atT ≈ Φ Teq , i.e. its density todayρmc ≈ 140Φ3 (1 + Φ)ρ̄a (Teq )The height of the plot is cut at Φ = 20.

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Axion stars

E.Kolb & IT, Phys.Rev. D50 (1994) 769

18 / 30

Models: QCD axions
τgr ∼109 yr
Φ4



Mc
10−13 M

2 

3

m
26 µeV-Φ ∼ 1 ⇒ τgr ∼ 109 yr
Mass fraction in miniclusters with Φ > Φ0
E.Kolb & IT, Phys.Rev. D49 (1994) 5040

Φ ∼ 103 ⇒ τgr ∼ hr

Universe filled with Bose stars!
Abundance? Mass function?
Thorough study of axion production from from topological defects is needed
M. Kawasaki, K. Saikawa, T. Sekiguchi, Phys.Rev. D91 (2015) 065014
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Models: QCD axions
τgr ∼109 yrΦ4Mc10−13 M2 3m
26 µeV-Φ ∼ 1 ⇒ τgr ∼ 109 yr
Mass fraction in miniclusters with Φ > Φ0
E.Kolb & IT, Phys.Rev. D49 (1994) 5040

Φ ∼ 103 ⇒ τgr ∼ hr

Universe filled with Bose stars!
Abundance? Mass function?
Thorough study of axion production from from topological defects is needed
M. Kawasaki, K. Saikawa, T. Sekiguchi, Phys.Rev. D91 (2015) 065014
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ALP Bose star instability

V (a) = m2 fa2

θ2−2

θ44!

+ ...

M̃
Self-coupling of axions is negative
and axion Bose stars are unstableagainst collapse.

4
20I. Tkachevω̃c0Axion stars

ω̃
50

100

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ALP Bose star collapse, stage I
Self-similar wave collapse
1015t̃j = −3.3 · 10−4
−1.9 · 10−7−8.2 · 10−111ρ̃|r̃ψ̃|2/y 21010|χ∗|2/y 210−4105

Critical star10−8110−610−310−21102

1/2106y ≡ r̃/|t̃|r̃Black hole does not form for fa < MP l
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Collapse, stage II: Relativistic Bosenova
Repeated explosions
Field evolution in the center

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Radial profiles at different times

Axion stars

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Collapse, stage II: Relativistic Bosenova
Repeated explosions
Field evolution in the center

Radial profiles at different times

ρ(t, 0)/m2f 2103

1
Eq. (16)

10−3
1800

mt

ρ(t, 0)/m2f 2103

2700

1

10−30
5 · 103bc d104

mt

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Collapse, stage II: Relativistic Bosenova
Repeated explosions
Field evolution in the center

Radial profiles at different times

ρ/m2 f 2

1
Eq. (16)

10−3
1800

mt

1

10−3

ρ/m2 f 2

ρ(t, 0)/m2f 2103

2700

ρ/m2 f 2

ρ(t, 0)/m2f 2103

05 · 103
bc d

104

10−2

self-sim

ilar

10−4

mt = 802110−2

waves →

10−4
1

selfsimilar

10−2
10−4

Axion stars

mt = 2253
slow waves

400mt

I. Tkachev

mt = 01200

mr400
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Decay of Bose star on relativistic axions
Total emitted energy fraction

Spectra of emitted particles

10∆Erel /∆E2f /MP2 l = 10−7−8100.3

const

0.1

310

∆E/McrdE/dk × (m/f )2/N∗

1050

I. Tkachev

3k/m

6linear

0.6

0.3
0

9f /MplAxion stars

3 · 10−4

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Phenomenological implications (QCD axions)Less diffuse DM -> smaller signals in DM detectorts
But rare strong signals during encounters with debris of tidally
disrupted miniclusters
P.Tinyakov, IT and K. Zioutas, JCAP 1601 (2016) 035

Gravitational microlensing and femtolensing
E.Kolb & IT, Astrophys.J 460 (1996) L25
M.Fairbairn, et. al, PRL 119 (2017) 021101

Decay of Bose stars
Decay to relativistic self
Resolution of tension between low and high z observations?
Z.Berezhiani, A.Dolgov & IT, Phys.Rev. D 92 (2015) 061303

Decay to radiophotons
Relation to FRB?
IT, JETP Letters 101 (2015) 1
А.Iwazaki, PRD 91(2015) 023008

Relation to ARCADE 2 excess and/or anomalous 21 cm signal?
J.Kehayias, T.Kephart & T.Weiler, JCAP 1510 (2015) 053
S.Fraser et al, arXiv:1803.03245
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Fast Radio Bursts and axion Bose-stars
FRB - mysterious astrophysical phenomena

Short radio flash, 1 ms

Radius of axion Bose-star 1 ms

Cosmological origin, z ∼ 1Energy release 1038 − 1040 ergs

Minicuster mass
10−12 M = 2 × 1042 ergs

Huge brightness temperature
TB ∼ 1036 K

Bose-star can explode in a burst ofcoherent radiation

Rate: ∼ 104 events/day for thewhole sky.

We have 1024 miniclusters just in aGalaxy

Can FRBs be explained by axion star explosions into pure radiation?
IT, JETP Letters 101 (2015) 1
А. Iwazaki, PRD 91(2015) 023008

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Fast Radio Bursts and axion Bose-stars
FRB - mysterious astrophysical phenomena

Short radio flash, 1 ms

Radius of axion Bose-star 1 ms

Cosmological origin, z ∼ 1
Energy release 1038 − 1040 ergs

Minicuster mass
10−12 M = 2 × 1042 ergs

Huge brightness temperature
TB ∼ 1036 K

Bose-star can explode in a burst ofcoherent radiation

Rate: ∼ 104 events/day for thewhole sky.

We have 1024 miniclusters just in aGalaxy

Can FRBs be explained by axion star explosions into pure radiation?
IT, JETP Letters 101 (2015) 1
А. Iwazaki, PRD 91(2015) 023008

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Fast Radio Bursts and axion Bose-stars
FRB - mysterious astrophysical phenomena

Short radio flash, 1 ms

Radius of axion Bose-star 1 ms

Cosmological origin, z ∼ 1
Energy release 1038 − 1040 ergs

Minicuster mass
10−12 M = 2 × 1042 ergs

Huge brightness temperature
TB ∼ 1036 KBose-star can explode in a burst ofcoherent radiation

Rate: ∼ 104 events/day for thewhole sky.

We have 1024 miniclusters just in aGalaxy

Can FRBs be explained by axion star explosions into pure radiation?
IT, JETP Letters 101 (2015) 1
А. Iwazaki, PRD 91(2015) 023008

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Axion direct detection
Minicluster abundance
Typical miniclusters with Φ ≈ 1:1024 in the Galaxy
1010 pc−3 in the Solar neighborhood
Minicluster radius ∼ 107 km
Direct encounter with the Earth once in 105 years
During encounter density increases by a factor 108
for about a dayBut, some miniclusters are destroyed in encounters with stars.
This may change the prospects for DM detection.

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Axion direct detection
Tidal streams from miniclusters
Probability of a minicluster disruption
 n Φ−3/2 (1 + Φ)−1/2
100
Just disk crossings. No actual orbits integration.
P (Φ) = 0.022

P.Tinyakov, IT and K. Zioutas, JCAP 1601 (2016) 035

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Axion direct detection
Crossing tidal streams from miniclusters
Mean number of encounters with axion streams producing amplification factor larger
than A, as a function of A. Twenty year observation interval is assumed.

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P.Tinyakov, IT and K. Zioutas, JCAP 1601 (2016) 035
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Axion direct detection
Crossing tidal streams from miniclusters
Simulation of expected PWS in cavity experiments
Jan 2026
Jan 2025
Jan 2024
Jan 2023
Jan 2022
Jan 2021
Jan 2020
0.1

Jan 2018

P − PN [10−22 W]

Jan 2019

0.05

Jan 2017
0Jan 201601aν−m2π [kHz]2C. O’Hare and A. Green, Phys.Rev. D95 (2017) 063017
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Conclusions

Bose condensation by gravitational interactions is very effcient
Large fraction of axion dark matter may consist of Bose stars
Phenomenological implications of Bose star existence are reach anddeserve further studies

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