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.

When the axion mass turns on, at QCD time,

1T

1t

1 1 GeVT 9

11

1) 3 10 eV(ap t

t

71 2 10 sect

Axion production by vacuum realignment

2 2 21 1 1 1

1

1

2

1

2( ) ( ) ( ) ( )a a at t t t

tn m a f

GeVT GeVT

V

a

V

a

1

0

3 7

60 1( ) ( )a aa a

R

Rt tm mn

initialmisalignmentangle

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

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

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

12

3

4

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 on the predicted time scale?

Thermalization occurs due to gravitational interactions

at time 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 1v

m t

3 3( ) / ( ) ( )g t H t t a t

In the linear regime, within the horizon, the axion BEC remains in the same state

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 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

an elliptic umbilic catastrophe

D-4

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

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)

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

10 arcmin = 2.2 kpc

29.2 kpc15.410.3

The caustic ring halo model assumes

• net overall rotation

• axial symmetry

• self-similarity

L. Duffy & PSPRD78 (2008)063508

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 =

in case of irrotational flow

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

in case of net overall rotation

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

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?

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 may enter into thermal contact with the axions by gravitational interactions as well

a

a

p

p

O. Erken, PS, H.Tam and Q. Yang, PRL 108 (2012) 061304

p may be any species of particle

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 numberof neutrinos

Effective number of neutrinos

WMAP 7 year:

J. Hamann et al. (SDSS):

Atacama Cosmology Telescope:

we will see …