Effective gravitational interactions of dark matter axions MASAHIDE YAMAGUCHI (Tokyo Institute of Technology) 11/29/13@New Perspectives on Cosmology, APCTP arXiv:1210.7080, PRD87, 085010 (2012) Ken’ichi Saikawa and MY arXiv:1310.0167, Toshifumi Noumi, Ken’ichi Saikawa, Ryosuke Sato, and MY

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

Introduction

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

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

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 ?

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)

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

Axion as dark matter

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

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)

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 ?

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.

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.

BEC of axion

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.

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.

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.

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 :

(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.

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.)

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)

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

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.

Effective gravitational interactions

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.

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.

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.

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

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.

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.

Evolution of axion toward BEC

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

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).

Thermalization rate Axions go toward BEC

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.

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

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)

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

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.

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)

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.

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.

Full Hamiltonian of axion - graviton system