Bose-Einstein Condensationof Dark Matter Axions

Pierre Sikivie

Texas Cosmology Network MeetingAustin, October 29-30, 2009

The Dark Matter is Axions

based on arXiv: 0901.1106with Qiaoli Yang

I’ll argue:

How so?

• axions differ from ordinary CDM because they form a BEC, and the axion BEC continues to rethermalize, tracking the lowest energy state.

• axion BEC appears to solve two puzzles with CDM 1) evidence for caustic rings of dark matter implies that the dark matter falls onto galactic halos with ``net overall rotation", implying whereas CDM predicts ,

2) unexpected alignments among CMBR multipoles .

0v! # r r

0v =! r r

Outline• Review of axion properties

• Cold dark matter axions form a rethermalizing Bose-Einstein condensate

• Axion BEC differs from ordinary CDM

• Compare CDM and axion BEC density perturbations withobservations in three arenas

1. upon entering the horizon 2. in the linear regime within the horizon 3. in the non-linear regime

CDMaxionBEC

f f

a

a

!!

610 GeVeVa

af

m

6

= 0.97 in KSVZ model 0.36 in DFSZ model

g!

5fa f f

a

L i g f ff

!" =

a

a

L g E Bf

!! !

" #

$ = %

ur ur

The remaining axion window

laboratory searches

510

1510

1010 (GeV)af

(eV)am1

510

! 1010

!

stellar evolution

cosmology

There are two axionpopulations: hot and cold.

When the axion mass turns on, at QCD time,

1T

1t

11 GeVT

7

12 10 sect

! "

9

1

1

1) 3 10 eV(ap t

t

! = "

Axion production by vacuum realignment

GeVT ! 1 GeVT ! 1

V

a

V

a

initialmisalignmentangle

2 2 2

1 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

5347

31

3 12

( )4 10( )

cm 10 GeV ( )

af a tn t

a t

! "# ! " $ %$ %

& ' & '

1

1

( )1

( )v( )

a

a t

m t a tt!

83

361

1234

3

(2 )( ) 10

10 GeV( v)

a

a

fn t

m!

!

"

# $ % &

' (

N

ifdecoupled

The cold axions thermalize andtherefore form a BEC

2

2

0 4 5

6

cm10 eV

m m

f!

"#105

#

1 $ % = 1.5 &10 ' (

64 ) *

a

a a

a

2

4

2...a

m

f!

1 +

4! = ... + L

0( ) v( ) ( )n t t H t! " << but …

At high phase space density

D. Semikozand I. Tkachev,1995, 1997

2

scatt 0vn ! "# N

relax 0vn ! "# N

scattering rate

thermalization rate

61( 10 ) N

relax 1 1(t H t! ( ) )

A critical aspect of axion BECphenomenology is whether the BECcontinues to thermalize after it has formed.

Axion BEC means that almost all axions goto one state.

However, only if the BEC continuallyrethermalizes does the axion state trackthe lowest energy state.

After the thermalization ratedue to self-interactions is

at time

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

2

H

n m!

!

! "#

# / $(1)

However, the thermalization ratedue to gravitational interactions

at time1t

-1with v)l m!= (

1 ( )( ) / ( ) ( )g a tt H t t a t !" # #

2

3

12

2 2

10 GeV

g

gf

H

G n m l

!8

" # / 4 $10 % &

' (

)

)

Gravitational interactions thermalize theaxions and cause them to form a BECwhen the photon temperature

After that

12

12eV10 GeV

fT!

" #100 $ %

& '

1v

m t!

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

" #

axion BEC

Except for a tiny fraction, all axions are in thesame state

2( ) [ ] ( ) ( )D x g x m xDµ µ! "µ µ ! µ! "# # # = $ $ % & $ =

†( ) [ ( ) ( ) ]x a x a x! ! ! ! !" #= $ % %+

0

†1| (

!) | 0N

N

N

a> = >

( 0)! =

N is the number ofaxions

in a homogeneousand isotropicspace-time.

0 3

2( )

im tA

a t

e!

" =

0 0 0 0

2

0 0 0 0

|

( )]

| [N T

g m

N Nµ! µ ! ! µ

"µ! "

# #

# #

< $ % $ + $ % $

+ &% $ $ & $ $

> = % %

%

In Minkowski space-time

Let

then

Let

then

0 ( ) ( )imtx e x

!" = #

21

2t

im

! " ## = $for non -relativisticmotion

( , )1( , ) ( , )

2

i x tx t B x t

mN

e!

" =rr r

2

00 ( , ) ( , ) ( , )( )T x t x t m B x t!" =r r r

hence

2

0vj j jT B !" = # $% #

1v( , ) = ( , )x t x t

m!"

rr r r

2

2

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! !" #+ $ =

ur ur

( )v v vt q! + "# = $#ur ur ur ur ur

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 t

kk t H k t G k t

m a! ! " # !

$ %& + & ' ' =( )

* +

r r r

( )

1 11 -5 292 4

2 144J

10 eV 10 g/cc16 1.02 10 cmG m

m! "

"

#$ % $ %= & ' ( ' (

) * ) *=l

In the linear regime within thehorizon, axion BEC and CDM areindistinguishable on all scalesof observational interest,

but

axion BEC differs from CDMin the non-linear regime &upon entering the horizon

CMBmultipoles

arealigned

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 upon entering the horizon.

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 structure ofspherically symmetric halos

(from Binney and Tremaine’s book)

Phase space structure ofspherically 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 angularmomentum (Tkachev, Wang + PS), the causticrings were predicted to be

in the galactic plane with radii

was expected for the Milky Wayhalo from the effect of angular momentumon 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 uponthe 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 starsH. 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 distanceappears circular, actually seen forscale height of order 1 kpcvelocity dispersion of order 20 km/s

may be caused by the n = 2 caustic ring ofdark 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! =ur r

neighboringprotogalaxy

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

! ( = ( + "#

!

= $#%(

r r rr r r r r r

r r

0v =! " r r

0v =! " r r

Tidal torque theory with axion BEC

Net overall rotation is produced because, in thelowest energy state, all axions fall with the sameangular momentum

v 0!" #ur r

For axion BEC

is minimized for given

when .

2

1 2

N

i

i

LE

I=

= !

1

N

i

i

L L

=

= !

1 2 3...

NL L L L= = = =

is allowed through theappearance of vortices; see discussionby Tanja Rindler-Daller and Paul Shapiroat this meeting.

v 0!" #ur r

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.