Bose-Einstein Condensationof Dark Matter Axions

Pierre Sikivie

IDM 2010

Montpellier

The Dark Matter is Axions

based on PRL 103 (2009) 111301 with Qiaoli Yang,

and arXiv: 1003.2426

I'll argue

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

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

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

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

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

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

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

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

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

Rotation curve of Andromeda Galaxyfrom L. Chemin, C. Carignan & T. Foster, arXiv: 0909.3846

thanks to S. BoyarskyandO. Ruchaiskiy

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

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

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

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

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.