1. Effective gravitational interactions of dark matter axions
MASAHIDE YAMAGUCHI
(Tokyo Institute of Technology)
Perspectives on Cosmology, APCTP
arXiv: , PRD87, (2012) Ken’ichi Saikawa and MY
arXiv: , Toshifumi Noumi, Ken’ichi Saikawa, Ryosuke Sato, and MY

2. Contents Introduction Bose-Einstein condensation of axion
What is an axion ?
　　　Axion dark matter, misalignment
Bose-Einstein condensation of axion
In-in formalism
Coherent state
Effective gravitational interactions
Evolution of axion toward BEC
Conclusions and discussion

3. Introduction

4. Evidence of dark matter
Flat rotation curve
of a spiral galaxy
Gravitational lensing
CMB …
Bergstrom 2000
HST

5. Candidates of dark matter
WIMPs
neutralino,
gravitino, axino,
sterile neutrino …
Axion
PBH …
Thermal production
(Kolb & Turner)
What is an axion ?

6. Strong CP problem θ vacuum : CP even CP odd
(allθs （０≦θ＜２π） are equivalent)
CP even
CP odd
Neutron electric dipole moment :
(CP odd)
Theoretical prediction (θvacuum) :
Experimental constraint :
(Baker et al. 2006)
Despite the equivalence of all θs , why is θ so small ?

7. Peccei-Quinn mechanism
Try to make θ dynamical :
Introduce a U(1) symmetry called Peccei-Quinn symmetry,
which is spontaneously broken at some scale.
Under U(1)PQ
NG boson a (axion) :
Assign PQ charges to quarks, which leads to
U(1)PQ-SU(3)c-SU(3)c anomaly.
const.
(A : constant associated with charges)
The observable θ parameter is determined by
(Fa = η/ A : axion decay constant)

8. Peccei-Quinn mechanism II
Through QCD effects, the axion acquires its mass
around the QCD scale ΛQCD ~ 100 MeV.
dynamically goes to the zero.

9. Axion as dark matter

10. Axion as a suitable candidate for DM
The couplings with other matter are suppressed by
the decay constant Fa ( ~ 109 – 1012 GeV).
Non-thermal production mechanisms
Emissions from strings (associated with U(1)PQ breaking)
and string-wall system (associated with QCD transition)
Misalignment mechanism
coherent oscillation
behaves like non-relativistic matter

11. Properties of axion dark matter
Axion starts its coherent oscillation at m(T1)=3H(T1),
Small velocity dispersion:
suitable for CDM (δv < 10-8)
Large occupation number:
( For WIMP, )
Bose-Einstein condensation ???
(Erken et al. 2012)

12. BEC of axion dark matter
Bose-Einstein condensation (BEC):
Large fraction of bosons are in the same (lowest) energy state
Critical temperature is much higher than the energy dispersion,
Necessary conditions :
Particles are bosons.
Number is conserved.
Large occupation number
Thermal equilibrium
Axions (coherent oscillations) satisfy the above three conditions.
How about the last condition ?

13. Thermalization Thermal equilibrium could be reached if
How to evaluate the interaction rate ?
WIMP (including sterile neutrino) :
Classical limit : while fixed
(classical particle limit)
classical points particle
Can use Boltzmann equation
Axion (coherent oscillations) :
Classical limit : while fixed
(classical field limit)
classical field
Cannot use Boltzmann equation
Need to derive the evolution equation.

14. Another subtlety in transitions
Consider transitions between different quantum states.
Particle kinetic regime :
Applies to almost all physical
systems including WIMP.
exchanged energy
Interaction rate
Condensed regime :
Applies to coherently oscillating axions.
The energy difference is too small to conclude that a particle has made
a transition from one highly occupied state to another in time Γ-1.
The transition makes sense if ,
when N particles made a transition from one state to another.

15. BEC of axion

16. Do axions form BEC ? Our approach Erken, Sikivie, Tam, Yang (2012) :
Describe axions as a set of quantum-mechanical oscillators (quantum operators).
Derive the evolution equation of each oscillator.
However,
The thermalization rate is estimated by comparing order of magnitudes
of quantum operators.
The thermalization is confirmed only by numerical calculations of toy model.
The photons also have thermal contact with axions, which drops photon
temperature and leads to large effective d.o.f. of neutrino (Neff ~ 6.77).
Our approach
We use the in-in formalism instead of treating operators directly,
which is suitable for following time development of the expectation value.
We use coherent states to describe such condensed axions.
If , axions would form BEC.

17. In-in formalism
In-in formalism is suitable for following time development of
the expectation value of some operator.
operators in
interaction picture
(upper suffix I will be omitted)
: initial (in) state given at the initial time t0.

18. Coherent state As an initial state, we would like to take a state
which represents the coherent oscillations of axions.
coherent state (characterized by a complex number αi) :
Expectation value of φ:
(inside the horizon )
(classical oscillation trajectory)
Mean square deviation of φ:
(vacuum fluctuations)
The coherent state has the same trajectory with
the classical field and the same fluctuation with the vacuum.

19. Initial state
In fact, the initial amplitudes of the oscillations fluctuate.
If the PQ phase transition occurs after inflation,
it can be different for each QCD horizon.
Even if the PQ phase transition occurs before inflation,
there are fluctuations originating from quantum fluctuations.
The coherently oscillating axions have momenta comparable to or less than
the Hubble scale at the time of the QCD phase transition.
The ``zero mode" is not exactly a single mode with zero momentum,
but the collection of plural modes near the ground state.
Initial state :

20. (Self-)interaction terms
We consider the following two types of interactions :
Self-coupling :
Gravitational coupling :
Here we dropped the processes which violate axion number.
Because such processes are forbidden due to the conservation of energy
and three momenta at the first order in HI as long as axions are non-relativistic.

21. Evolution of occupation number
For p < K (condensed mode),
In condensed regime , this first order term is relevant.
Otherwise (in particle kinetic regime), it rapidly oscillates and vanishes.
For p > K (particle-like mode),
go into second order
(We recover the usual Boltzmann equation.)

22. Thermalization rate
Thermalization rate is given by the time scale of the change
of the occupation number for condensed modes.
Self-coupling :
Gravitational coupling :
(Newtonian approximation)

23. Thermalization rate II
Axions go toward BEC
Is the Newtonian approximation justified ???

24. Interaction with other species
The interaction Hamiltonian with other species b :
Initial state :
(Here we assume b particles are in number states)
For p < K (condensed mode),
For p > K (particle-like mode),
Axions do not have thermal contact with other species at least at first order.
(This result is robust even if we consider axion-two photons couplings (a+bb).)
No significant effects on cosmological parameters.

25. Effective gravitational interactions

26. Effective gravitational interactions of axions
In general relativistic framework, the gravitational interactions
are mediated by metric perturbations.
e.g.
In the RD era, the Universe is almost homogeneous and isotropic,
and only time translational invariance is broken.
Similar to inflationary phase.
Effective field theory approach is useful.

27. Basic idea of effective field theory
(Cheung et al. 2008)
ρrad(t) spontaneously breaks time diffeomorphism inv.
Time-dependent spatial diffeo is unbroken.
In the low energy effective theory, any term
respecting the unbroken symmetry is allowed.
We can investigate the properties of perturbations generated
during RD without resort to a particular Lagrangian.

28. Axion - Graviton system
1. Radiations are dominant, so the effects of axions on background are neglected.
2. Time translation is broken, which leaves the time dependent spatial diffeo.
3. We take the Unitary gauge (δρrad = 0).
ADM decomposition:
Solving constraints to remove N and Ni yields the action for
dynamical fields, ζ, γ, φ.

29. Hamiltonian of axion - graviton system
In the interested region ma ≫ H, k/R,
( is subdominant.)

30. In-in formalism
In-in formalism is suitable for following time development of
the expectation value of some operator.
operators in
interaction picture
(upper suffix I will be omitted)
: initial (in) state given at the initial time t0.

31. Effective Hamiltonian for gravitational interactions of axions
At the leading order and the tree level,
We assume that the operator OI(t) does not contain the creation/annihilation operators of ζ and the in-state |in> vanishes by the annihilation operator of ζ.
The Newtonian approximation is reproduced when the momentum
transfer is subhorizon scale csk / (RH) ≫ 1,
On the other hand, it is significantly suppressed for a superhorizon scale.

32. Evolution of axion toward BEC

33. Evolution of occupation number
< > denotes the expectation value for the coherent states,
(tq : QCD phase transition)
coherent state :

34. Evolution of occupation number II
We decompose each mode into sub/super horizon parts
1. All external lines attached to the quartic coupling are
subhorizon sized (k1, k2, k3, p > kH).
2. One or two external lines are superhorizon sized, while the
other external lines and the momentum transfer |k1 − p|
are subhorizon sized (k1, k2 > kH and k3, p < kH, etc.).
we are particularly interested in the case 2, since it represents a transition from (into) modes
with higher momentum (subhorizon sized waves) into (from) almost homogeneous modes (superhorizon sized waves),
which is relevant to the transition into (from) BEC.
This case is relevant to
the transition into(from) BEC.
3. All external lines are superhorizon sized (k1, k2, k3, p < kH).

35. Thermalization rate
Axions go toward BEC

36. Conclusions and discussion
We have formulated the method to evaluate the
interaction rate of the coherently oscillating axions.
The in-in formalism and coherent states are used.
We derived the effective gravitational interactions for axions and
estimated the interaction rates among sub/super horizon modes.
Our results may suggest that axions form BEC at TBEC ~ keV.
The interactions with other species vanish at least at the
leading order.  no confliction with the standard cosmology
We have not yet confirmed the appearance of BEC directly.
We need to follow the development of the state instead of the
expectation value, and to take into account the thermal effects.
Seek for observable effects like caustics.

37. Unitary(Comoving) gauge
ρrad(t)
Unitary(Comoving) gauge : xi
Time slice (t = const. hypersurface) coincides with
ρrad= const. hypersurface. =0
The radiation perturbation is eaten by the metric.
The graviton (metric) has three degree of freedom:
Curvature perturbation ζ
Tensor perturbations γij

38. Action in unitary gauge (single component case)
Any quantities respecting the time-dependent diffeo. inv.
4-dim scalar
generic function of t, f(t)
　 in unitary gauge, which allows
any tensor with 0 upper index (g00 , R00, …)
Extrinsic curvature :
(All covariant derivatives of nμ can be written using Kμν and derivatives of g00)
: unit vector orthogonal to t =const.
All covariant derivatives of n_mu can be written using K_munu and derivatives of g^oo.
: projection tensor to t =const.
Note that
(3)R is redundant.
(all indices are contracted)

39. Expanding around FLRW background
Fluctuations around FLRW background:
0th and 1st order in fluctuations:
(Terms such as can be absorbed into
the above terms by integration by parts)
Variation w.r.t. g00 & gij

40. Expanding around FLRW background II
This is the most general action in unitary gauge ( ),
which satisfies the time-dependent spatial diffeo. invariance.
Note, however, that time diffeo. :
is broken.

41. Schematics of interaction processes
in-out
in-in
particle
particle
usual a+b→a+b process
1st order a+b→a+b (vanish)
in-in
in-in
condensed
condensed
particle
particle
condensed
condensed
particle
particle
1st order a+a→a+a (non-zero)
2nd order b+b→b+b (non-zero)

42. Caustics WIMP : Velocity field is irrotational
because it comes from gravitational forces
proportional to the gradient of Newton potential.
Inner caustics of dark matter halo is tent-like.
BEC axions :
Fall into the lowest energy state.
Velocity field becomes rotational
because each particle carries an equal amount of
angular momentum.
Inner caustics of dark matter halo is ring-like.
Recent observations suggest ring-like caustics.

43. Interaction rates among superhorizon modes
All external lines are superhorizon sized (k1, k2, k3, p < kH).
This result can be understood as evolution of the background field,
Effective action for φcl :
Integrating out ζ
The BG can evolve, but no transition
among superhorion modes.

44. Full Hamiltonian of axion - graviton system