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Bose-Einstein Condensationof Dark Matter Axions
Pierre Sikivie
COSMO/CosPA 2010 Conference
The University of Tokyo
September 27 - October 1, 2010
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The Dark Matter is Axions
based on PRL 103 (2009) 111301 with Qiaoli Yang,
and arXiv: 1003.2426
I'll argue
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Outline
• A brief review of axion physics
• Cold dark matter axions form a Bose-Einstein condensate
• The inner caustics of galactic halos
• Tidal torque theory with ordinary CDM and with axion BEC
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The Strong CP Problem
Because the strong interactions conserve P and CP, .
2
QCD 2...
32a ag
L G G
1010
The Standard Model does not provide a reason for to be so tiny,
but a relatively small modification of the model does provide a reason …
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If a symmetry is assumed,
relaxes to zero,
PQU (1)
and a light neutral pseudoscalar particle is predicted: the axion.
2
2
1... .
32 2a a
aa
gL G G
f
af
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f f
a
a
610 GeVeV
aa fm
= 0.97 in KSVZ model 0.36 in DFSZ model
g
5fa f fa
L i g f ff
aa
L g E Bf
����������������������������
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The remaining axion window
laboratory searches
510 15101010 (GeV)af
(eV)am 1 510 1010
stellar evolution
cosmology
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There are two axion populations: hot and cold.
When the axion mass turns on, at QCD time,
1T
1t
1 1 GeVT 71 2 10 sect
91
1
1) 3 10 eV(ap t
t
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Axion production by vacuum realignment
GeVT GeVT
V
a
V
a
initialmisalignmentangle
2 2 21 1 1 1
1
1
2
1
2( ) ( ) ( ) ( )a a at t t t
tn m f
1
0
3 7
60 1( ) ( )a aa a
at t
am mn
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Cold axion properties
• number density
• velocity dispersion
• phase space density
5 347 31
3 12
( )4 10( )
cm 10 GeV ( )af a t
n ta t
1
1
( )1
( )v( )
a
a t
m t a tt
83 3
6112
34
3
(2 )( ) 10
10 GeV( v)
a
a
fn t
m
N
ifdecoupled
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Bose-Einstein Condensation
if identical bosonic particles
are highly condensed in phase space
and their total number is conserved
and they thermalize
then most of them go to the lowest energy
available state
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why do they do that?
by yielding their energy to the
non-condensed particles, the
total entropy is increased.
BECpreBEC
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A critical aspect of axion BEC phenomenology is whether the BEC continues to thermalize after it has formed.
Axion BEC means that almost all axions go to one state.
However, only if the BEC continually rethermalizes does the axion state track
the lowest energy available state.
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Thermalization occurs due to gravitational interactions
at time
2
3
12
2 2
10 GeV
g
f
G n m l
1t
-1with v)l m
1 ( )( ) / ( ) ( )g a tt H t t a t
2
2
Gm
q
q
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Gravitational interactions thermalize the axions and cause them to form a BEC when the photon temperature
After that
1
2
12eV
10 GeV
fT
1v
m t
3 3( ) / ( ) ( )g t H t t a t
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In the linear regime, within the horizon,
axion BEC density perturbations obey
Jeans’ length
42
0 2 4( , ) 2 ( , ) 4 ( , ) 0
4t tk
k t H k t G k tm a
1 1
1 -5 292 42 144
J10 eV 10 g/cc
16 1.02 10 cmG mm
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In the linear regime within the
horizon, axion BEC and CDM are
indistinguishable on all scales
of observational interest,
but
axion BEC differs from CDM
in the non-linear regime &
upon entering the horizon
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x
x.
DM particles in phase space
DM forms caustics in the non-linear regime
x
xx
.x
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Phase space distribution of DM in a homogeneous universe
z
z.
ztHz )(v
-1210v
for WIMPs
-17v 10 -8v 10
for axions (preBEC)
for sterile neutrinos
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The dark matter particles lie on a 3-dimensional sheet in
6-dimensional phase space
the physical density is the projection of the phase space sheet onto position space ( , t) = t) ( , t)r r rv v
����������������������������������������������������������������������
z
z.
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The cold dark matter particles lie on a 3-dimensional sheet in
6-dimensional phase space
the physical density is the projection of the phase space sheet onto position space ( , t) = t) ( , t)r r rv v
����������������������������������������������������������������������
z
z.
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Phase space structure of spherically symmetric halos
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(from Binney and Tremaine’s book)
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Phase space structure of spherically symmetric halos
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Galactic halos have inner caustics as well as outer caustics.
If the initial velocity field is dominated by net overall rotation, the inner caustic is a ‘tricusp ring’.
If the initial velocity field is irrotational, the inner caustic has a ‘tent-like’ structure.
(Arvind Natarajan and PS, 2005).
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simulations by Arvind Natarajan
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The caustic ring cross-section
an elliptic umbilic catastrophe
D-4
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Galactic halos have inner caustics as well as outer caustics.
If the initial velocity field is dominated by net overall rotation, the inner caustic is a ‘tricusp ring’.
If the initial velocity field is irrotational, the inner caustic has a ‘tent-like’ structure.
(Arvind Natarajan and PS, 2005).
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On the basis of the self-similar infall model(Filmore and Goldreich, Bertschinger) with angular momentum (Tkachev, Wang + PS), the caustic rings were predicted to be
in the galactic plane
with radii
was expected for the Milky Way halo from the effect of angular momentum on the inner rotation curve.
1,2,3...n
maxj 0.18
rot max40kpc v j
220km/s 0.18n
na
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Effect of a caustic ring of dark matter upon the galactic rotation curve
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Composite rotation curve(W. Kinney and PS, astro-ph/9906049)
• combining data on
32 well measured
extended external
rotation curves
• scaled to our own galaxy
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Inner Galactic rotation curveInner Galactic rotation curve
from Massachusetts-Stony Brook North Galactic Pane CO Survey (Clemens, 1985)
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(l , b) (80 , 0 ) o o 10 10o o
IRAS m
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(l , b) (80 , 0 ) o o 10 10o o
IRAS m
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Outer Galactic rotation curve
R.P. Olling and M.R. Merrifield, MNRAS 311 (2000) 361
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Monoceros Ring of stars
H. Newberg et al. 2002; B. Yanny et al., 2003; R.A. Ibata et al., 2003; H.J. Rocha-Pinto et al, 2003; J.D. Crane et al., 2003; N.F. Martin et al., 2005
in the Galactic planeat galactocentric distance appears circular, actually seen forscale height of order 1 kpcvelocity dispersion of order 20 km/s
may be caused by the n = 2 caustic ring of dark matter (A. Natarajan and P.S. ’07)
20 kpcr 0 0100 270l
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Rotation curve of Andromeda Galaxyfrom L. Chemin, C. Carignan & T. Foster, arXiv: 0909.3846
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10 arcmin = 2.2 kpc
29.2 kpc15.410.3
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from L. Duffy and PS, Phys. Rev. D78 (2008) 063508
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The caustic ring halo model assumes
• net overall rotation
• axial symmetry
• self-similarity
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The specific angular momentum distribution on the turnaround sphere
a
2
m xˆ( , ) ˆ ˆˆ( )( )
nR t
n tt
z nj
2 2
3 9( )R t t
0.25 0.35
Is it plausible in the context of tidal torque theory?
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Tidal torque theory
neighboringprotogalaxy
Stromberg 1934; Hoyle 1947; Peebles 1969, 1971
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Magnitude of angular momentum
fits perfectly ( )0.25 0.35
0.05G. Efstathiou et al. 1979, 1987
max 0.18j
from caustic rings
1
2
5
2
max| | 16 8
5 3 10 3
L E
G M
j
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Tidal torque theorywith ordinary CDM
neighboringprotogalaxy
the velocity field remains irrotational
v 0 ����������������������������
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For collisionless particles
If initially,
then for ever after.
v v, ) , ) v( , ) v( , )
, )
dr t r t r t r t
dt t
r t
0v =
0v =
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Tidal torque theorywith axion BEC
v 0 ����������������������������
net overall rotation is obtained because, in the lowest energy state,all axions fall with the same angular momentum
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For axion BEC
is minimized for given
when .
2
1 2
Ni
i
LE
I
1
N
ii
L L
1 2 3 ... NL L L L
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Tidal torque theorywith axion BEC
v 0 ����������������������������
net overall rotation is obtained because, in the lowest energy state,all axions fall with the same angular momentum
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Self-Similarity
a comoving volume
3
( )
( ) ( , ) ( ( , )V t
t d r r t r r t
( )r a t x ( ) , ) ( )r a t x t x
( )
( , )( , )
t
r tr t
( ) , ) ( ) ( )r a t x t a t x
4 30( ) ( ) ( ) ( ) ( ( ))x
V
t t a t d x x x x
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Self-Similarity (yes!)
time-independent axis of rotation
2
3ˆ ˆ( ) ( )t z a t z t
5
3ˆ( )L t z t
1 4 523 9 3( )
ˆ( , )R t
n t t tt
0.33provided
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Conclusions
Axions differ from ordinary CDM because they form
a Bose-Einstein condensate
Axion BEC and CDM are indistinguishable in the linear
regime inside the horizon on all scales of observational
interest.
If the dark matter is axions, the phase space structure
of galactic halos predicted by tidal torque theory is
precisely, and in all respects, that of the caustic ring
model proposed earlier on the basis of observations.