Bose-Einstein Condensationof Dark Matter Axions

Pierre Sikivie (U. of Florida)

Center for Particle Astrophysics

Fermilab, August 6, 2009

The Dark Matter is Axions

based on arXiv: 0901.1106

with Qiaoli Yang

I’ll argue:

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

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

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 …

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

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

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

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.

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

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

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

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

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

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

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

We have

and

with

( v) 0t ����������������������������

( )v v vt q ����������������������������������������������������������������������

2

2

1

2mq

To recover CDM, let m go to infinity

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

CMBmultipoles

are aligned

quadrupole:

octupole: M. Tegmark, A. de Oliveira-CostaA. Hamilton, 2003

C. CopiD. HutererD. SchwarzG. Starkman, 2006

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.

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

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)

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

from L. Duffy and PS, Phys. Rev. D78 (2008) 063508

Tidal torque theory with CDM

The velocity field remains irrotational

v 0 ����������������������������

neighboringprotogalaxy

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 ����������������������������

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.

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.

- 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

Hierarchical clustering introduces effective velocity dispersion

effv

effv 30 km/s

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

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