Probing ultralight axion dark matter with gravitational-wave detectors

Probing ultralight axion dark matter with gravitational-wave detectors Arata Aoki (Kobe Univ.) with Jiro Soda (Kobe Univ.) A.A. and J. Soda, Phys. Rev. D 93 (2016) 083503. A.A. and J. Soda, arxiv:1608.05933. 2016.12.14 @ Kyoto Univ.

Contents Ultralight Axion Dark Matter Probing Axion DM with Pulsar Timing Array [Khmelnitsky & Rubakov, JCAP 1402(2014)019] Probing Axion DM with Laser Interferometers [Aoki & Soda, arxiv:1608.05933] Axion in f(r) theory [Aoki & Soda, PRD 93(2016)083503] Summary

Why Ultralight Axion Dark Matter? [ 1 / 25] Small-scale problems with standard CDM. CDM predicts overabundance of sub-galactic structures. e.g.) core-cusp problem, etc. An ultralight scalar field (axion) DM resolves these issues because of effective quantum pressure. db = 2 mv ' 38 kpc 10 22 ev m 10 3 v. No evidence of supersymmetry (neutralinos).

Ultralight Axion DM [ 2 / 25] Model : S = S EH + Z d 4 x p g apple 1 2 (@ )2 1 2 m2 2. Self-interaction and interactions with SM particles are suppressed by a large decay constant. f a Occupation number of axions in DM halo : N x 3 p 3 dm m(mv) 3 1086 dm 0.3 GeV/cm 3 10 22 ev m 4. (v 10 3 = 300 km/s) A classical field description is quite good.

Axion Oscillation [ 3 / 25] Solution of EoM (Klein-Gordon eq.) : (t, ~x) = 0 (~x) cos[mt + (~x)]. : monochromatic frequency E = p m 2 + k 2 ' m k. (10 kpc) 1 m 10 22 ev (0.1 pc) 1 Energy density : dm = 1 2 2 + 1 2 m2 2 ' 1 2 m2 2 0 ' 0.3 GeV/cm 3. Pressure : p dm = 1 2 1 2 2 m2 2 ' dm cos(2mt +2 ). oscillating

Oscillating Gravitational Potentials (1/2) [ 4 / 25] The oscillating pressure induce the oscillation of the gravitational potentials. Metric : ds 2 = (1 + 2 )dt 2 +(1 2 ) ij dx i dx j. Trace of the Einstein equation : R = T. 6 +2r 2 (2 )= 0 [1 + 3 cos(2mt)]. The oscillating part of satisfies 6 =3 0 cos(2mt).

Oscillating Gravitational Potentials (2/2) [ 5 / 25] Oscillating part of the gravitational potential : = dm 8m 2 cos(2mt). Amplitude & Frequency : dm =5 10 18 8m2 dm 0.3 GeV/cm 3 f = 2m 2 =5 10 8 Hz m 10 22 ev 10 22 ev m. 2. This oscillating gravitational potentials can be detected by gravitational-wave detectors.

Pulsar Timing Array Experiments (1/2) [ 6 / 25] The oscillating gravitational potential induces an time delay in arrival time of pulsar signals, with angular frequency 2m and amplitude t = dm 8m 3 sin md + (~x) (~x p ). ~x : position of the detector, ~x p : position of the pulsar. D = ~x ~x p. This can be compared with residual due to GWs. h c =2 p 3 =2 10 15 dm 0.3 GeV/cm 3 10 23 2 ev. m [Khmelnitsky & Rubakov (2014)]

Pulsar Timing Array Experiments (2/2) [ 7 / 25] 2.3 10 23 ev m ev [Khmelnitsky & Rubakov (2014)] 10 12 10 23 10 22 10 21 hc 10 13 10 14 10 15 CURRENT LIMIT 5 yrs of PPTA 20 pulsars 100 ns 10 yrs of SKA 100 pulsars 50 ns 10 16 Scalar Field Dark Matter 10 17 10 9 10 8 10 7 10 6 observed frequency Hz

Signal on Laser Interferometers (1/3) [ 8 / 25] The solar system is moving through the DM halo with a velocity v 10 3 = 300 km/s. Hence, we feel the wind of axion! Since the axion wind looks like scalar gravitational waves, we can detect it by gravitational-wave laser interferometers. ˆn axion wind ˆm ~v

Signal on Laser Interferometers (2/3) [ 9 / 25] Metric perturbations : g ij = 2 A ij +2B,ij. Axion wind produces B,ij = dm 8m 2 v iv j cos(2mt). A cannot be detected by interferometers since it is an isotropic mode. Detector signal : s = D ij g ij, D ij 1 2 (ˆm i ˆm j ˆn iˆn j ).

Signal on Laser Interferometers (3/3) [10 / 25] Detector signal : s(t) = dmv 2 8m 2 cos(2mt). (ˆv ˆm) 2 (ˆv ˆn) 2 Signal amplitude & frequency : dm v 2 8m 2 =4.8 10 24 v 10 3 2 10 22 ev m 2. f 2m 2 =5 10 8 Hz m 10 22 ev.

Signal and Sensitivity of planned detectors [11 / 25] 10 22 ev 10 23 10 21 m [ev] 10 19 10 17 10 15 10 13 [Aoki & Soda (2016)] ASTROD-GW elisa strain DECIGO signal f [Hz]

Axion in f(r) theory (1/2) [12 / 25] Since we focus on the gravitational interaction, the signal should be sensitive to the theory of gravity. The dark energy problem might be resolved by modified gravity theories. Hence, we should discuss the axion oscillation in f(r) theory, which is the simplest modified gravity. Z S = 1 2 d 4 x p g [R + f(r)] + S m. f(r) R, f R f 0 (R) 1

Axion in f(r) theory (2/2) [13 / 25] Field equation : 1 G µ 2 g µ f +(R µ + g µ r µ r )f R = T µ. Trace of the field equation : 3 f R + Rf R 2f R = T. This can be seen as the equation of motion for, f R i.e. f R is a dynamical degree of freedom. When f R, f and f R ' f R 1 R, we obtain 3 f R + R = dm [1 + 3 cos(2mt)].

Gravitational Potentials in f(r) theory (1/2) [14 / 25] 3 f R + R = dm [1 + 3 cos(2mt)]. The long-time average of the field equation gives R 0 hri = dm. Ricci scalar : R = 6 +2r 2 (2 ). The time-dependent part of the Ricci scalar is R R R 0 = R dm ' 6. Using the field equation, satisfies 2 = dm cos(2mt) fr.

Gravitational Potentials in f(r) theory (2/2) [15 / 25] 2 = dm cos(2mt) fr. The oscillating part of the gravitational potential : = dm 8m 2 cos(2mt)+1 2 f R hf R i. In f(r) theory, the oscillating gravitational potential is directly related to the extra dynamical degree of freedom f R. In order to see this more clearly, we introduce a scalar field ' f R (scalaron).

Axion in scalar-tensor theory (1/2) [16 / 25] Field equation : 3 f R + R = dm [1 + 3 cos(2mt)]. The equation of motion : ' + V e,' = dm cos(2mt). The effective potential is defined as V e,' = 1 3 A(') dm, ' = f 0 (A). Formula for the gravitational potential : = dm 8m 2 cos(2mt)+1 2 ' h'i.

Axion in scalar-tensor theory (2/2) [17 / 25] In order to obtain the time-dependent part of the gravitational potential, all we have to do is 1. calculating the effective potential, V e,' = 1 3 A(') dm, ' = f 0 (A). 2. solving the equation of motion, ' + V e,' = dm cos(2mt). 3. and substituting the solution into = dm 8m 2 cos(2mt)+1 2 ' h'i.

R 2 model (1/2) [18 / 25] We first consider the simplest model : f(r) = R2 6M 2. Effective potential : V e = 1 2 M 2 (' ' 0 ) 2, ' 0 f 0 ( dm )= dm 3M 2 Equation of motion : Solution : ' + M 2 (' ' 0 )= dm cos(2mt). ' = ' 0 + dm M 2 (2m) 2 cos(2mt).

R 2 model (2/2) [19 / 25] Oscillating gravitational potential : = E 1 (2m/M) 2 cos(2mt), E dm 8m 2 (a) M 2m : M 2m (b) : ' E cos(2mt). ' M 2m E cos(2mt) E. (c) M 2m : resonance!

Exponential model (1/4) [20 / 25] Exponential model : f(r) = R2 0 3 2 M 2 exp apple R 1 R 0. The solar-system test requires ' 0 f 0 ( dm ) = dm 3 M 2. 10 15. 2 10 22 ev 2. & 3 10 4 2m M m The model can pass the solar-system test even if M 2m 10 22 ev.

Exponential model (2/4) Effective potential : V e (') = 0 3 ' apple 1 ln ' ' 0. [21 / 25] V e Curvature singularity (R! +1) 0 '/' 0 1 e '/' 0 Maximum amplitude : max ' ' ' 0. 10 15.

Exponential model (3/4) [22 / 25] Equation of motion : nonlinear forced oscillator. dm ' ' ln = dm cos(2mt). 3 Resonance curve : ' 0 = 10 3 R 2 model Exp. model log 10 ( / E) max ' (2m/M) 2

Exponential model (4/4) [23 / 25] New resonances at M/2m ' n =1, 2, 3,... amp. n =3 n =2 n =1 0 0.1 0.2 (2m/M) 2 0.3 0.9 1.0 1.1 1.2 All the details are in Mechanics (Landau & Lifshitz)

Signal and Sensitivity of planned detectors [24 / 25] m = 10 22 ev M = 10 18 ev 10 15 ev ASTROD-GW elisa strain DECIGO v 2 max 10 21 solar-system test f [Hz]

Summary [25 / 25] An ultralight axion with mass around candidate of DM. 10 22 ev is a The oscillating gravitational potential induced by the axion oscillating can, in principle, be detected by future gravitational-wave detectors. It is sensitive to models of gravity. Remarkably, a resonance could amplify the signal in f(r) theory. In the nonlinear case, there are many resonance points, and the signal could be detected by planned interferometers.