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Thermal boson starsL. Arturo Ureña-López!
Departamento de Física, Universidad de Guanajuato in collaboration with Argelia Bernal & Alberto Díez
!1
Motivation• Boson stars are self-gravitating configurations of bosons at zero
temperature.
• They are usually called self-gravitating Bose-Einstein condensates
• This would also mean that:
• Self-gravitating configurations must exist, which would be in thermal equilibrium, and that should be able to condensate at low enough temperatures.
• Were first studied in the seminal paper by Ruffini & Bonazzola, PR (1967)
• Relativistic: Einstein-Klein-Gordon equations (complex field)
• Non-relativistic: Schrödinger-Poisson equations
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Boson Stars (Newtonian approach)
• Equations of motion for equilibrium configurations:
!
!
• Normalization of the wavefunctions:
• Mass density:
• Particle number:
• Conservation of the total particle number:
Z| |2dV = 1
NT = N0
N0
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r2 = 2(mU � ✏)
r2U = N0| |2
⇢ = N0| |2
Thermal Boson Stars (Newtonian approach)
• Equations of motion for equilibrium configurations:
!
!
• Normalization of the wavefunctions:
• Mass density:
• Particle numbers:
• Conservation of the total particle number:
Z j
⇤l dV = �jl
⇢ =X
j=0
Nj | |2j
Nj =1
e�(✏j�µ) � 1
NT =
X
j
Nj = const.
!4
r2 j = 2(mU � ✏j) j
r2U =X
j=0
Nj | |2j
Multi- state configurations!Matos & U-L, GRG (2007) Bernal et al, PRD (2010)
U-L & Bernal, PRD (2010)
What must we expect?• All states are initially populated, and
their population is uniquely determined by the temperature T, the total particle number N, and the trapping potential.
• The energy levels are fixed by the trapping potential, and remain fixed all throughout.
• There is a critical temperature below which the ground state is massively populated.
• This is exactly the famous Bose-Einstein condensation
!5
High T ———-> Low TBEC, Wikipedia
What must we expect?
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Bose-Einstein condensation of an ideal gasThermodynamic limit
Chemical potential for ρph > ζ(3/2):
βµ = −0
Number of particles for ρph > ζ(3/2):
N = N0 + Vλ3
Tζ(3/2)
Number of zero-energy particles forρph > ζ(3/2):
N0 = N − V(
mkBT2π!2
)3/2ζ(3/2)
0 2 4 6 8 10phase-space density
-1
-0.8
-0.6
-0.4
-0.2
0
βµ
uniform1µm3 box
0 0.2 0.4 0.6 0.8 1 1.2 1.4temperature [µK]
0
20
40
60
80
100
N 0
ζ(3/2)
Cold gases 2009 – p.5Finite particle number effects !vs standard textbook approx.
• All states are initially populated, and their population is uniquely determined by the temperature T, the total particle number N, and the trapping potential.
• The energy levels are fixed by the trapping potential, and remain fixed all throughout.
• There is a critical temperature below which the ground state is massively populated.
• This is exactly the famous Bose-Einstein condensation
Thorsten Köhler, DPA, University College London
What did we find?• All states are initially populated, and
their population is uniquely determined by the temperature T, the total particle number N, and the trapping gravitational potential.
• The energy levels are not fixed by the trapping gravitational potential, and do not remain fixed afterwards.
• There is a critical temperature below which the ground state is massively populated.
• This is exactly the gravitational Bose-Einstein condensation
!7
Fugacity vs the temperature, !for different number of states
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
0 1 2 3 4 5 6 7 8 9 10
z =
e`(µ
< ¡
0)
T/Tc
567
What did we find?
!8
Particle number vs the temperature, !for different number of states
• All states are initially populated, and their population is uniquely determined by the temperature T, the total particle number N, and the trapping gravitational potential.
• The energy levels are not fixed by the trapping gravitational potential, and do not remain fixed afterwards.
• There is a critical temperature below which the ground state is massively populated.
• This is exactly the gravitational Bose-Einstein condensation
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
0 1 2 3 4 5 6 7 8 9 10
Nj/N
T/Tc
N0/NN1/NN2/NN3/NN4/NN5/N
What did we find?
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Particle number vs the temperature, !for different number of states
0
5
10
15
20
25
0 5 10 15 20 25 30
l r2
r
T << TcT = TcT >> Tc
• All states are initially populated, and their population is uniquely determined by the temperature T, the total particle number N, and the trapping gravitational potential.
• The energy levels are not fixed by the trapping gravitational potential, and do not remain fixed afterwards.
• There is a critical temperature below which the ground state is massively populated.
• This is exactly the gravitational Bose-Einstein condensation
Conclusions• We have found, for the first time, numerical solutions for the BEC
transition of a bunch of self-gravitating bosons.
• There are strong indications of a sharp phase transition for the formation of the BEC.
• Interestingly enough, the critical temperature does depend only upon the total particle number.
!
• There are indications that equilibrium configurations must be stable.
• Leftover: Would it be useful to describe rotation curves in scalar field galaxy halos?
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Low & High Temperatures
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• Low:!
!
!
• Chemical potential:
!
• Ground-state configuration!
• Scaling properties:
✏0 � µ ' T/N
r2 j = 2(mU � ✏j) j
r2U = N | 0|2
0
5
10
15
20
25
0 5 10 15 20 25 30l
r2
r
T << TcT = TcT >> Tc
Ground-state boson stars! are stable!
Low & High Temperatures
!12
• High:!
!
!
• Chemical potential:
!
• Multi-state configuration!
• Scaling properties:
r2 j
= 2(mU � ✏j
) j
r2U = (N/jmax
+ 1) ·j
maxX
0
| j
|2
0
5
10
15
20
25
0 5 10 15 20 25 30l
r2
r
T << TcT = TcT >> Tc
Multi-state boson stars! are stable if excited states!
are less populated than !the ground state
What is the critical temperature?
• The equations of motion have the same scaling properties at high & low temperatures.
!
• This means in particular that:
• … but this property should also hold for the critical temperature. From our numerical experiments we find that:
