Self-gravitating system made of
axions
Juan Barranco
Argelia Bernal
The Invisible Universe, Paris 2009
Self-gravitating system made of axions – p. 1
Plan of the talk
1. Motivation
2. Boson stars
• Generalities
• Self-interaction
3. Towards a realistic axion star
4. Axion Star
5. Discussion and conclusions
Self-gravitating system made of axions – p. 2
1.Motivation
Gµν = 8πGTµν
Self-gravitating system made of axions – p. 3
1. Motivation
M. Taoso, G. Bertone and A. Masiero, arXiv:0711.4996 [astro-ph]
The axion is still a good candidate!!
Self-gravitating system made of axions – p. 4
1. Motivation
The axion
• Real scalar field
• Its properties have been constrained:
1010 GeV ≤ fa ≤ 1012 GeV
10−5 eV ≤ m ≤ 10−3 eV
• At late stages of the evolution of the universe it acquires an
non-vanishing potential energy density
φ
2 2
V (φ) = m fa 1 − cos
fa
Self-gravitating system made of axions – p. 5
2. Boson stars
Boson stars (BS) are gravitationally bounded systems made of scalar particles.
Two different approaches:
•
•
A complex, classical scalar field
D.J. Kaup, Klein-Gordon Geon Phys. Rev.172 , 1331
Quantized real scalar fields
R. Ruffini and S. Bonazzola, Systems of selfgravitating particles in general relativity and
the concept of an equation of state, Phys. Rev. 187, 1767 (1969).
Both are equivalent (at least without self-interactions)
A small review:
Gµν
Tµν =⇒ T̂µν =⇒ hQ|T̂µν |Qi
«
„
dV
φ̂ = 0 ,
= 8πG < T̂µν > ,
2−
dφ̂2
”
X“
−
+
+iEn t
−iEn t
l∗
l
+ µlmn Rnl (r)Ym (θ, ψ)e
µlmn Rnl (r)Ym (θ, ψ)e
φ → φ̂ =
lmn
with the commutation relations
−
−
+
[µ+
lmn , µl′ m′ n′ ] = [µlmn , µl′ m′ n′ ] = 0
,
−
[µ+
lmn , µl′ m′ n′ ] = δll′ δmm′ δnn′ .
Self-gravitating system made of axions – p. 6
2. Boson stars
•
|Qi = |N, 0, 0, 0i is an state for which all the N particles are in the ground state
The ground state l = m = 0 (real since φ̂ = φ̂† )
−iE1 t
+iE1 t
φ̂ = µ+
+ µ−
100 R10 (r)e
100 R10 (r)e
hQ|T00 |Qi
=
hQ|T11 |Qi
=
′′
+
Rln
„
B′
»„ 2
«
′2 –
E1
R
1 2
2
~ N
+ m2 R10
+ 10
−
2m
B
A
»„ 2
«
′2 –
E1
R
1 2
2
~ N
− m2 R10
+ 10
2m
B
A
A′
2
+
−
r
2B
2A
«
′
+A
Rnl
"
2
Enl
B
− m2 −
#
l(l + 1)
Rln = 0
r2 A
Quantized real scalar fields yields to Boson stars and not to oscillatons
Self-gravitating system made of axions – p. 7
2. Boson stars
«
′2 –
σ
1
+ 1 σ2 +
−
B
A
«
„
« »„
′2 –
1
1
σ
B′
1
− 2 1−
− 1 σ2 +
−
ABx
x
A
B
A
« –
»„
„
′
′ «
B
A
1
1
+
−
−1 σ
σ′ + A
σ ′′ +
x
2B
2A
B
1
A′
+
A2 x
x2
definitions x = rm, R10 =
„
1
1−
A
√1
σ,
4πG
«
»„
=
0,
=
0,
=
0
B = E12 B
Boundary Conditions
•
•
•
0.06
0.05
σ (0)=0.05
σ (x)
0.04
0.03
0.02
Flat geometry at infinity
Arbitrary σ(0)
»
– 2
mpl
1
1
M = xmax 1 −
2
A(xmax ) m
In the Newtonian limit ρ(x) = |σ(x)|2
0.01
0
0
Regular at origin
10
20
x
30
40
50
Self-gravitating system made of axions – p. 8
2. Boson stars: self-interaction
M. Colpi, S. L. Shapiro and I. Wasserman, Phys. Rev. Lett. 57 (1986) 2485.
Tµν =
1
1
λ
(∂µ Φ∂ν Φ∗ + ∂µ Φ∗ ∂ν Φ) − gµν (g αβ ∂α Φ∂β Φ∗ + m2 |Φ|2 + |Φ|4 ) .
2
2
2
Λ=
λMp2
4πm2
«
′2 –
σ
1
Λ
+ 1 σ2 + σ4 +
−
B
2
A
«
„
« »„
′2 –
Λ
σ
B′
1
1
1
− 2 1−
− 1 σ2 − σ4 +
−
ABx
x
A
B
2
A
«
«
»„
„
–
′
′
B
A
1
1
+
−
− 1 σ − Λσ 3
σ′ + A
σ ′′ +
x
2B
2A
B
A′
1
+
A2 x
x2
„
1
1−
A
«
»„
=
0,
=
0,
=
0
Self-gravitating system made of axions – p. 9
2. Boson stars: self-interaction
Mmax
2
M
p
≈ 0.22Λ1/2
m
Inclusion of the potential can drastically change the Boson Stars
characteristics
Self-gravitating system made of axions – p. 10
3. Towards an axion star
V (φ) =
m2 fa2
1
1
V (φ) ∼ m2 φ2 −
2
4!
1 − cos
m
fa
φ
fa
1 m2 6
φ +
φ − ...
4
6! fa
4
V (φ) → hQ|V (φ̂)|Qi
φ̂ = µ+ R(r)e−iE1 t + µ− R(r)e+iE1 t
µ|Qi = 0
hQ|φ̂2 |Qi = R2
hQ|φ̂4 |Qi = 2R4
hQ|φ̂6 |Qi = 5R6
Self-gravitating system made of axions – p. 11
4. Results
x = rm ,
A′
1
σ,
R= √
4πG
2
mp
1
Λ=
24π fa
E2
B → 2B
m
2
′2
Λ 6
1
σ
1
1
2 2Λ 4
+ 2 1−
−
+1 σ − σ +
+
σ
= 0,
2
A x x
A
B
2
A
2
′2
′
2
1
1
1
B
Λ 4 σ
Λ 6
2
− 2 1−
−
− 1 σ +2 σ +
−
σ
= 0,
ABx x
A
B
2
A
2
′
′
3 2 5
B
A
1
1
′′
′
3
σ +
+
−
σ +A
− 1 σ+2Λσ − Λ σ
= 0
x 2B 2A
B
2
Self-gravitating system made of axions – p. 12
4.Results
2
M[mp /ma]
0.15
0.1
Λ =40
Λ =60
Λ =80
Λ =100
0.05
0
0.01
0.02
0.03
σ(0)
0.04
0.05
0.07
0.06
Mass of a boson star as a function of the central value of the scalar field.
Self-gravitating system made of axions – p. 13
4. Results
150
0.014
0.012
R99
0.01
2M/R99
Λ =40
Λ=60
Λ=60
Λ=100
100
0.008
0.006
Λ =40
Λ =60
Λ =80
Λ =100
50
0.004
0.002
0
0
0.01
0.02
0.03
0.04
σ(0)
0.05
0.06
0
0.01
0.02
0.03
σ(0)
0.04
0.05
0.06
As Λ → ∞, 2M/R99 → 0: Newtonization
Self-gravitating system made of axions – p. 14
4. φ4 vs φ6
»
„ «–
φ
V (φ) = m2 fa2 1 − cos
fa
0.2
4
Up to φ
6
Up to φ
M[mp2/ma]
0.15
0.1
0.05
0
0
0.01
0.02
0.03
0.04
σ(0)
0.05
0.06
0.07
Self-gravitating system made of axions – p. 15
5. What about the actual axion values?
1010 GeV ≤ fa ≤ 1012 GeV
10−5 eV ≤ m ≤ 10−3 eV
1
Λ=
24π
„
mp
fa
«2
∼ 1013 − 1017
Self-gravitating system made of axions – p. 16
5. What about the actual axion values?
1010 GeV ≤ fa ≤ 1012 GeV
10−5 eV ≤ m ≤ 10−3 eV
1
Λ=
24π
„
mp
fa
«2
∼ 1013 − 1017
"„
„
«2
«
1
R′2
1
E2
m2 4
4π
2
+ 2 1−
+m
R +
+
−
− 2
A2 r
r
A
mp
B
4fa2
A
«
«
»„ 2
„
1
E
R′2
m2 4
4π
1
B′
2
2
− 2 1−
−m R +
R +
−
− 2
ABr
r
A
mp
B
4fa2
A
„
«
»„ 2
′
′ «
1
B
A
m2 3
E
′′
2
′
R +
+
−
−m R+ 2 R −
R +A
r
2B
2A
B
fa
A′
6
R
4
72fa
#
=
0,
m2 6
R
72fa4
–
=
0,
m2 5
R
24fa4
–
=
0
m2
Self-gravitating system made of axions – p. 16
5. What about the actual axion values?
1010 GeV ≤ fa ≤ 1012 GeV
10−5 eV ≤ m ≤ 10−3 eV
1
Λ=
24π
„
mp
fa
«2
∼ 1013 − 1017
"„
„
«2
«
1
R′2
1
E2
m2 4
4π
2
+ 2 1−
+m
R +
+
−
− 2
A2 r
r
A
mp
B
4fa2
A
«
«
»„ 2
„
1
E
R′2
m2 4
4π
1
B′
2
2
− 2 1−
−m R +
R +
−
− 2
ABr
r
A
mp
B
4fa2
A
„
«
»„ 2
′
′ «
1
B
A
m2 3
E
′′
2
′
R +
+
−
−m R+ 2 R −
R +A
r
2B
2A
B
fa
A′
fa
R = √ σ,
m
mp
r=
fa
r
m
x,
4π
6
R
4
72fa
#
=
0,
m2 6
R
72fa4
–
=
0,
m2 5
R
24fa4
–
=
0
m2
4πfa2
α= 2
mp m
Self-gravitating system made of axions – p. 16
4. Results
fa
R = √ σ,
m
mp
r=
fa
r
m
x,
4π
4πfa2
α= 2
mp m
A(x) = 1 − a(x)
«
4
′2
6–
1
mσ
σ
σ
+ 1 m2 σ 2 −
+α
+
B
4
(1 − a)
72
»„
«
4
′2
6–
1
mσ
σ
σ
aB
− (1 − a)Bx
− 1 m2 σ 2 +
+α
−
B′ +
x
B
4
(1 − a)
72
„
«
«
»„
′
′
3
5–
2
B
a
mσ
σ
1
σ ′′ +
+
+
− 1 m2 σ +
−
σ ′ + (1 − a)
x
2B
2(1 − a)
B
3
24
a(1 + a)
+ (1 − a)2 x
a′ +
x
»„
=
0,
=
0,
=
0
Self-gravitating system made of axions – p. 17
4. Results
0
0.00025
-5e-14
a(x)
σ(x)
0.0002
0.00015
-1e-13
0.0001
-1.5e-13
5e-05
0
0
500
1000
x
1500
-2e-13
0
2000
mp
r=
fa
r
500
1000
x
1500
2000
m
x
4π
for m = 10−5 eV r = 3.84−3 (meters)×x and
„
1
M = 4.83 × 1024 Kg 1 −
A
«
→ M = 2.4 × 1013 Kg ∼ 10−17 M⊙
Self-gravitating system made of axions – p. 18
Is it bad?
30
Mini
MACHOS
Forbidden
because of
dynamical
instability
Forbidden because of
MACHO Microlensing
25
log10 t/ys
Rich
Larg
20
ter
e Ga
rf G
15
10 15
laxy
Dwa
10
Clus
Forbidden
because of
arguments on
massive BHs
Mo
10 12
Mo
alax
y 1 8
0 M
o
t = 13.7 Gyr
Forbidden because
of gravothermal
instability
5
0
-8
-6
-4
-2
0
2
4
6
log10 m/Mo
X. Hernandez, T. Matos, R. A. Sussman and Y. Verbin,
Phys. Rev. D 70, 043537 (2004)
Self-gravitating system made of axions – p. 19
5. Conclusions
• We have constructed the self-gravitating system made of
axion-like particles. For the present values of ma and fa , they
should have very small masses and should be meter-size objects.
• implications?
• stability
Self-gravitating system made of axions – p. 20