Axion BEC Dark Matter
Pierre Sikivie
Vistas in Axion Physics
Seattle, April 22-26, 2012
Collaborators: Ozgur Erken, Heywood Tam, Qiaoli Yang
An argument why the dark matter is axions
1. Cold dark matter axions thermalize and form a
Bose-Einstein condensate.
2. The axion BEC rethermalizes sufficiently fast that
axions about to fall onto a galactic halo almost all
go to the lowest energy state for given total
angular momentum.
3. As a result the axions produce
- caustic rings of dark matter
- in the galactic plane
- with radii
4. There is observational evidence for the existence
of caustic rings of dark matter
- in the galactic plane
- with radii
- with overall size consistent with tidal torque
theory
5. The evidence for caustic rings is not
explained by other forms of dark matter.
Ordinary cold dark matter (WIMPs, sterile
neutrinos, non-rethermalizing BEC, …)
forms tent-like inner caustics.
There are two cosmic axion
populations: hot and cold.
T1
t1
When the axion mass turns on, at QCD time,
T1 1GeV
t1 2107 sec
1
pa (t1 )  3109 eV
t1
Axion production by vacuum realignment
V
V
a
a
TGeV
TGeV
1
na (t1 )  ma (t1 )a(t1 )    f a 2  (t1 )2
1
2
2t1
2
3
 R1 
a (t0 ) ma na (t1 )  ma
 R0 
7

6
initial
misalignment
angle
Cold axion properties
• number density
fa
4 10 

n(t ) 

 12

3
cm  10 GeV 
47
• velocity dispersion
• phase space density
5
3
 a(t1 ) 


 a(t ) 
1 a(t1 )
 v(t ) 

ma t1 a(t )
(2 )
if
decoupled
fa


N n(t )
 10  12

4
10

GeV


(ma  v)3
3
3
61
3
8
3
Cold axion properties
• number density
fa
4 10 

n(t ) 

 12

3
cm  10 GeV 
47
• velocity dispersion
• phase space density
5
3
 a(t1 ) 


 a(t ) 
1 a(t1 )
 v(t ) 

ma t1 a(t )
(2 )
if
decoupled
fa


N n(t )
 10  12

4
10

GeV


(ma  v)3
3
3
61
3
8
3
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.
preBEC
BEC
the axions thermalize and
form a BEC after a time
the axion fluid obeys
classical field equations
the axion fluid does not obey
classical field equations
the axion BEC rethermalizes
the axion fluid obeys
classical field equations
the axion fluid does not obey
classical field equations
Axion field dynamics
From
self-interactions
From gravitational self-interactions
In the “particle kinetic” regime
implies
When
4
1
2
3
D. Semikoz & I. Tkachev, PRD 55 (1997) 489D.
After
, axions thermalize in the
“condensed” regime
implies
for
and
for self-gravity
Toy model thermalizing in the
condensed regime:
with
i.e.
50 quanta among 5 states
316 251 system states
Start with
Number of particles
Total energy
Thermal averages
Integrate
Calculate
Do the
approach the
the predicted time scale?
on
Thermalization occurs due to
gravitational interactions
PS + Q. Yang, PRL 103 (2009) 111301
with l  m v)-1
q
Gm 2
q2
at time t1
g (t ) / H (t )ta(t ) a(t )
1
Gravitational interactions thermalize the
axions and cause them to form a BEC
when the photon temperature
After that
1
 v 
mt
g (t ) / H (t )t a(t )
3
3
In the linear regime, within the horizon, the
axion BEC remains in the same state
axion BEC density perturbations obey
4


k
2
t  (k , t )  2 H t (k , t )   4 G 0 
 (k , t )  0
2 4 
4m a 

Jeans’ length
J
 16  G  m
1

2
4

1
2
-5
29



10 eV
10 g/cc 
14
 1.02 10 cm 
 


 m  

1
4
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 when it
rethermalizes
in the non-linear regime &
upon entering the horizon
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
in case of net overall rotation
The caustic ring cross-section
D-4
an elliptic umbilic catastrophe
in case of irrotational flow
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  n  1,2,3...
40kpc  vrot  jmax 
an 



n  220km/s  0.18 
jmax  0.18 was expected for the Milky Way
halo from the effect of angular momentum
on the inner rotation curve.
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 curve
Inner Galactic rotation curve
from Massachusetts-Stony Brook North Galactic Pane CO Survey (Clemens, 1985)
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 plane
at galactocentric distance r 20kpc
appears circular, actually seen for 1000  l 2700
scale height of order 1 kpc
velocity dispersion of order 20 km/s
may be caused by the n = 2 caustic ring of
dark matter (A. Natarajan and P.S. ’07)
Rotation curve of Andromeda Galaxy
from L. Chemin, C. Carignan & T. Foster, arXiv: 0909.3846
10.3
10 arcmin = 2.2 kpc
15.4
29.2 kpc
The caustic ring halo model
assumes
L. Duffy & PS
PRD78 (2008)
063508
• net overall rotation
• axial symmetry
• self-similarity
The specific angular momentum
distribution on the turnaround sphere
R(t )
(nˆ, t ) jmax nˆ( zˆ nˆ )
t
2
R(t )t
2 2

3 9
0.25 0.35
Is it plausible in the context of
tidal torque theory?
Tidal torque theory
neighboring
protogalaxy
Stromberg 1934; Hoyle 1947; Peebles 1969, 1971
Tidal torque theory
with ordinary CDM
neighboring
protogalaxy
 v  0
the velocity field remains irrotational
For collisionless particles
d v
v
 r , t )  r , t ) v(r , t )  v(r , t )
dt
t
  r , t )

If

v=0 initially,
then
v=0 for ever after.
in case of irrotational flow
Tidal torque theory
with 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
N
2
Li
E 
i 1 2 I
is minimized for given
N
L Li
i 1
when
L1  L2  L3 ... LN
.
in case of net overall rotation
The specific angular momentum
distribution on the turnaround sphere
R(t )
(nˆ, t ) jmax nˆ( zˆ nˆ )
t
2
R(t )t
2 2

3 9
0.25 0.35
Is it plausible in the context of
tidal torque theory?
Tidal torque theory
with 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
1

L| E | 2
5

GM 2
0.05
G. Efstathiou et al. 1979, 1987
6
8
1

  jmax
5  3 103 
jmax  0.18
from caustic rings
fits perfectly ( 0.25 0.35 )
The specific angular momentum
distribution on the turnaround sphere
R(t )
(nˆ, t ) jmax nˆ( zˆ nˆ )
t
2
R(t )t
2 2

3 9
0.25 0.35
Is it plausible in the context of
tidal torque theory?
Self-Similarity
3
d r (r , t )r ( (r , t )

V (t )
 (t )
a comoving volume
  r a(t )x,t ) ( x )
r a(t )x
 (r , t )
 (r , t )
  r a(t )x,t )a(t ) ( x )
  (t )
 (t ) 0 (t )a (t )  d x ( x )x( x  ( x ))
4
3
V
Self-Similarity (yes!)
 (t )zˆa(t )zˆt
L(t )zˆt
2
3
5
3
time-independent axis of rotation
2
R(t )
(nˆ, t )
t
t
1 4

3 9
provided
t
5
3
 0.33
Conclusion:
The dark matter looks like axions
Baryons and photons may enter into
thermal contact with the axions by
gravitational interactions as well
p
p may be any species
of particle
p
a
a
O. Erken, PS, H.Tam and Q. Yang, PRL 108 (2012) 061304
Relativistic axion modes and photons reach thermal
contact with the cold axions if the cold axion correlation
length reaches horizon size by the time of equality
If photons, baryons and axions all reach
the same temperature before decoupling
photons cool
baryon to photon
ratio
effective number
of neutrinos
Effective number of neutrinos
WMAP 7 year:
J. Hamann et al. (SDSS):
Atacama Cosmology Telescope:
we will see …