bose-einstein condensation of dark matter axions pierre sikivie (u. of florida) center for particle...
TRANSCRIPT
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Bose-Einstein Condensationof Dark Matter Axions
Pierre Sikivie (U. of Florida)
Center for Particle Astrophysics
Fermilab, August 6, 2009
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The Dark Matter is Axions
based on arXiv: 0901.1106
with Qiaoli Yang
I’ll argue:
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Outline• Review of axion properties
• Cold dark matter axions form a Bose-Einstein condensate
• Axion BEC differs from ordinary CDM
• Compare CDM and axion BEC density perturbations with observations in three arenas
1. upon entering the horizon 2. in the linear regime within the horizon 3. in the non-linear regime
CDMaxionBEC
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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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The cold axions thermalize and therefore form a BEC
22
0 4 5
6
cm10 eV
m m
f
a
a a
a
24
2...a
m
f
L
0( ) v( ) ( )n t t H t but …
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At high phase space density
D. Semikoz and I. Tkachev, 1995, 1997
2scatt 0 vn N
relax 0 vn N
scattering rate
thermalization rate
61( 10 )N
relax 1 1(t H t
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More generally, axion-like particles (ALPs) form a BEC
without relationship between and
ALP oscillations start at
ALP number density
ALP velocity dispersion
f m
11
mt
12( )t f mn
v
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ALP phase space density
ALP self-coupling strength
ALP scattering cross-section
Hence ALP thermalization rate
1 1 1 1 0 1( ) ( ) 2 ( ) v( )
(1)
/t H t t n t t
O
N
2 2
3 2
6
( v)
n f
m m
N
2m
f
0 2m
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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 state.
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After the thermalization rate due to self-interactions is
at time
2n m
3 1( )( ) / ( ) ( ) a tt H t t a t
Self-interactions are insufficient to rethermalize axion BEC after t1even if they cause axion BEC at t1.
1t
1t
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However, the thermalization rate 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
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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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axion BEC
Except for a tiny fraction, all axions are in the same state
2( ) [ ] ( ) ( )D x g x m xD
†( ) [ ( ) ( ) ]x a x a x
0†1
| (!
) | 0NNN
a
( 0)
N is the number ofaxions
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in a homogeneous and isotropic space-time.
0 3
2( )
im tA
a t
e
0 0 0 0
20 0 0 0
|
( )]
| [N T
g m
N N
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In Minkowski space-time
Let
then
Let
then
0 ( ) ( )imtx e x
21
2ti m
for non -relativisticmotion
( , )1( , ) ( , )
2i x tx t B x t
mNe
200 ( , ) ( , ) ( , )( )T x t x t m B x t
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hence
20 vj j jT B
1v( , ) = ( , )x t x t
m
22
1 1
4v v ( )j jjk k k jkT
m
stresses related to the Heisenberg uncertaintyprinciple tend to homogenize the axion BEC
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We have
and
with
( v) 0t ����������������������������
( )v v vt q ����������������������������������������������������������������������
2
2
1
2mq
To recover CDM, let m go to infinity
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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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CMBmultipoles
are aligned
quadrupole:
octupole: M. Tegmark, A. de Oliveira-CostaA. Hamilton, 2003
C. CopiD. HutererD. SchwarzG. Starkman, 2006
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Upon entering the horizon
CDM density perturbations evolve linearly
the density perturbations in the axion BEC
evolve non-linearly because the axionBEC
rethermalizes
axion BEC provides a possible mechanism for
the alignment of CMBR multipoles through the
ISW effect.
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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
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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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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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from L. Duffy and PS, Phys. Rev. D78 (2008) 063508
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Tidal torque theory with CDM
The velocity field remains irrotational
v 0 ����������������������������
neighboringprotogalaxy
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Tidal torque theory with axion BEC
Net overall rotation is produced because, in the lowest energy state, all axions fall with the same angular momentum
v 0 ����������������������������
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Summary
• axion BEC and CDM are indistinguishable in the
linear regime inside the horizon on all scales of
observational interest.
• axion BEC may provide a mechanism for net
overall rotation in galactic halos.
• axion BEC may provide a mechanism for the
alignment of CMBR multipoles.
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Implications:
1. At every point in physical space, the distribution of velocities is discrete, each velocity corresponding to a particular flow
at that location .
2. At some locations in physical space, where the number of flows changes, there is a caustic, i.e. the density of dark matter is very high there.
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- the number of flows at our location in the Milky Way halo is of order 100
- small subhalos from hierarchical structure formation produce an effective velocity dispersion
but do not destroy the sheet structure in phase space
- the known inhomogeneities in the distribution of matter are insufficient to diffuse the flows by gravitational scattering
- present N-body simulations do not have enough particles to resolve all the flows and caustics (see however: Melott and Shandarin, Stiff and Widrow, Shirokov and Bertschinger, and more recently: White and Vogelsberger, Diemand and Kuhlen.)
effv 30 km/s
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Hierarchical clustering introduces effective velocity dispersion
effv
effv 30 km/s
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The Big Flow
• density
• velocity
• velocity dispersion
24 35 1.7 10 gr/cmd
previous estimates of the total local halo density range from 0.5 to 0.75 10 gr/cm-24 3
in the direction of galactic rotation
in the direction away from the galactic center
5v m/s
r
km/s)ˆ100ˆ470(v5 r
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Experimental implications
• for dark matter axion searches
- peaks in the energy spectrum of microwave photons
from conversion in the cavity detector
- high resolution analysis of the signal yields a more sensitive search (L. Duffy and ADMX collab.)
• for dark matter WIMP searches
- plateaux in the recoil energy spectrum from elastic WIMP collisions with target nuclei
- the total flux is largest around December
(Vergados; Green; Gelmini and Gondolo; Ling, Wick &PS)
a