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Bose-Einstein Condensationof Dark Matter Axions
Pierre Sikivie
ASK2011 Conference
Seoul, Korea
April 11, 2011
Outline
• Dark matter
• Axions
• Bose-Einstein Condensation
• Caustics
Fritz Zwicky
1898 - 1974
COMA, a cluster of galaxies
COMA cluster of galaxies
Virial Theoremcluster mass
cluster radius
velocities of galaxies in the cluster
Using the virial theorem, Zwicky (1933) showed that the Coma cluster contains roughly 100 times more mass than accounted for by the luminous matter in the cluster.
2v GMR
Vera Rubin and Kent Ford
Galactic rotation curves
2 ( )v ( )
G M rr
r
r
galactic mass
rotation speed
Rubin and Ford (1970) found that the rotation speed of M31 is constant at large radii implying that the total galactic mass increases proportionately to the radius.
Galaxies are surrounded by halos of dark matter.
the bullet cluster (D. Clowe, A. Gonzales and M. Markevitch. 2004)
WMAP
Concurring data from: Boomerang, Maxima, South Pole, Saskatoon, DASI, VSA, …
Cosmological Parameters
• Age of the universe 13.7 Gigayear
• Spatial curvature none
• Energy density 0.95 10 gr/cm
• Energy fractions vacuum energy atoms and molecules dark matter
-29 3
0.740.04b0.22dm
What is thedark matter ?
The candidates
• the axion
• the WIMP
• the sterile neutrino
• others
17v 10a c
1210vW c
8v 10 c
2eV/a cm
2100 GeV/W cm
210 keV/cm
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 …
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
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
����������������������������
The remaining axion window
laboratory searches
510 15101010 (GeV)af
(eV)am 1 510 1010
stellar evolution
cosmology
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
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
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
why do they do that?
by yielding their energy to the
non-condensed particles, the
total entropy is increased.
BECpreBEC
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
PS + Q. Yang, PRL 103 (2009) 111301
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
x
x.
DM particles in phase space
DM forms caustics in the non-linear regime
x
xx
.x
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
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.
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.
Phase space structure of spherically symmetric halos
(from Binney and Tremaine’s book)
Phase space structure of spherically symmetric halos
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).
simulations by Arvind Natarajan
The caustic ring cross-section
an elliptic umbilic catastrophe
D-4
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).
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
Effect of a caustic ring of dark matter upon the galactic rotation curve
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
Inner Galactic rotation curveInner Galactic rotation curve
from Massachusetts-Stony Brook North Galactic Pane CO Survey (Clemens, 1985)
(l , b) (80 , 0 ) o o 10 10o o
IRAS m
(l , b) (80 , 0 ) o o 10 10o o
IRAS m
Outer Galactic rotation curve
R.P. Olling and M.R. Merrifield, MNRAS 311 (2000) 361
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
Rubin and Ford (1970) found that the rotation speed of M31 is constant at large radii implying that the total galactic mass increases proportionately to the radius.
Galaxies are surrounded by halos of dark matter.
Rotation curve of Andromeda Galaxyfrom L. Chemin, C. Carignan & T. Foster, arXiv: 0909.3846
10 arcmin = 2.2 kpc
29.2 kpc15.410.3
The caustic ring halo model assumes
• net overall rotation
• axial symmetry
• self-similarity
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?
Tidal torque theory
neighboringprotogalaxy
Stromberg 1934; Hoyle 1947; Peebles 1969, 1971
Tidal torque theorywith ordinary CDM
neighboringprotogalaxy
the velocity field remains irrotational
v 0 ����������������������������
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 =
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
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
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?
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
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
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
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
Conclusion:
The dark matter looks like axions
Baryons and photons enter into thermal contact with the axion BEC
a
a
p
p
Ozgur Erken, PS, Heywood Tam and Qiaoli Yang - April 2011
Photons, baryons and axions all reach the same temperature before decoupling
photons cool
baryon to photon ratio
effective numberof neutrinos
we will see …
