Testing the axion-photon coupling with polarization measurements of Sagittarius A* Guan-Wen Yuan yuangw@pmo.ac.cn a Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences b School of Astronomy and Space Science, University of Science and Technology of China June 29, 2020
Outline ◮ Birefringence from Axion and Profile Density ◮ The Sgr A* Polarization Data and Constraint ◮ Discussion: DM Density and Wash-Out ◮ Summary
Birefringence from Axion This ALP field can interact with the electromagnetic field and give rise to a rotation effect on the photon polarization, which is called the birefringence effect. The relevant Lagrangian terms include L = − 1 4 F µν F µν + 1 + g a γ ∂ µ a ∂ µ a − m 2 a 2 � 4 aF µν ˜ F µν � (1) 2 Thus, the classical EoM and solution for axion field: � a + m 2 a = g a γ E · B (2) a ( t , x ) = a 0 ( x ) sin ( mt + δ ( x )) (3)
Birefringence from Axion Additionally, the EoM and solution for photons: ¨ a ∇ × A + ˙ A − ∇ 2 A = g a γ ( ˙ A × ∇ a ) (4) d 3 k � � (2 π ) 3 A p k e p (ˆ � k ) e ik · x − i dt ω p A ( t , x ) = (5) p = ± The dispersion relation of EoM and two polarization states propagate: ± − k 2 ∓ g a γ ( ˙ a + ˆ w 2 k · ∇ a ) | k | = 0 (6) � a + ˆ 1 ± g a γ ( ˙ k · ∇ a ) ≈ k ± 1 w ± = k 2 g a γ ∂ 0 a (7) k
Birefringence from Axion The difference between the frequencies of the two polarization components is translated into the change of the polarization angle for a linearly polarized emission. � t obs � t obs ∆ φ = 1 ( w + − w − ) dt = 1 2 g a γ ∂ 0 adt 2 t emit t emit (8) = 1 2 g a γ [ a ( t obs , x obs ) − a ( t emit , x emit )] Where the a ( t obs , x obs ) and a ( t emit , x emit ) are the ALP amplitude in observation and emission points, respectively. If we assumption ρ DM = 1 2 m 2 a 2 0 in Eq.(8) � 1 � � ρ DM 2 π t 2 ∆ φ ≃ 5 ◦ sin � T + δ ( x ) 2 × 10 9 GeV / cm 3 (9) � � � g a γ m � − 1 × 10 − 12 GeV − 1 10 − 18 eV
CDM Simulation A slice of density field of ψ DM simulation on various scales at z =1 1 , which can be described by soliton+NFW dark matter profile. 1 H.Schive,etc. Nature Physics, 2014, 10(7):496-499.
Soliton+NFW DM Profile For ultralight axion DM, there can generate a cored soliton, due to the balance between gravitational interaction and quantum pressure. The De Broglie wavelength of axion ( r c = 2 π mv ) can describe solitonic core radius, and the axion field is coherent and homogeneous in it.Thus 2 , � − 4 � − 2 � � m r c M ⊙ pc − 3 , 190 × for r < r c 10 − 18 eV 1 pc ρ DM = ρ 0 r / R g (1+ r / R g ) 2 , for r > r c (10) 2 H.Schive,etc. Physical review letters, 2014, 113(26):261302.
Very-Long Baseline Interferometry, VLBI Figure 1: Some stations of the EHT, which operating at 1.3mm wavelength by VLBI technique.
Baseline Figure 2: There are four telescopes have been used to observe Sgr A*, including CARMA, SMT,JCMT and SMT. This is the interferometric fractional polarization measurements for Sgr A*.
Polarization Data Figure 3: Polarization Data from M.Johnson Fig.S8 3 m ≃ 4 · 10 3 ( 10 − 18 eV The axion typical oscillation period: T = 2 π ) sec . m m up = 2 . 92 × 10 − 18 eV , r c = 0 . 018 pc , ρ DM = 7 . 46 × 10 9 GeV / cm − 3 m low = 2 . 90 × 10 − 19 eV , r c = 0 . 183 pc , ρ DM = 7 . 36 × 10 7 GeV / cm − 3 3 M.Johnson,etc, Science, 2015, 350(6265):1242-1245.
The Sgr A* Data Analysis there will be a constant angle φ 0 depending on our observation. Thus, the final polarization angle φ a is φ a = ∆ φ + φ 0 (11) The least square fitting is utilized in our work, and the χ 2 function is expressed as N ( φ obs − φ pre ) 2 χ 2 = � (12) σ 2 i i =1 where φ obs and σ i are the observed polarization angle and its dispersion of angle. The predicting polarization angles φ pre of the ALP model and the background model are φ a and φ 0 in Eq.(11), respectively.
Results Figure 4: Constraints for ALP DM in the plane m − g a γ at 95% CL.
Discussion I: DM Density In the center of the galaxy, ultra-light boson can from a soliton core with large density as Eq.(10). A super-massive black hole within the soliton core can further increase the density around it. For example, a Kerr black hole with large spin leads to superradiance of the boson when its Compton wavelength matches the gravitational radius of the black hole. The pile-up of the dark matter also will happen outside Schwarzschild black hole. In this case, one could imagine a huge enhancement of the density. This makes our benchmark halo model Eq.(10) a very conservative one.
Discussion II: Wash-Out by Resolution Figure 5: Strength and order of the polarization field from 1.3mm VLBI 4 . � δφ/ 2 1 cos( µ t + m φ ) d φ = sin( m δφ/ 2) cos µ t (13) δφ m δφ/ 2 − δφ/ 2 In the case where the observing disk is face-on, the spatial resolution of Sgr A* in this observation is about 0 . 5 π and the wash-out factor is 0.9, thus the washout effect is negligible. 4 M.Johnson,etc, Science, 2015, 350(6265):1242-1245.
Discussion III: Wash-Out by Line of Sight The radiation area observed, is within a disk of 10 times of the black hole Schwarzschild radii. When the axion mass is less than 9 . 65 × 10 − 18 eV, its Compton wavelength is much longer than this length scale. Assuming the source length in the line of sight direction is the same order, the average along this direction becomes again negligible.
Summary ⋆ Axions, with initial motivation to solve strong CP problem, are also an important ultra-light dark matter candidate. ⋆ Photon emitted from soliton core has birefringent effect making polarization angle oscillate. The amplitude of the oscillation is dependent on the axion density of the emission position. ⋆ Outside black hole, ultralight axion, which can be the most dense in universe, can generate a cored soliton. The de Broglie wavelength of axion can describe its radius. ⋆ Depending on EHT excellent spatial resolution, the polarization data of neighboring Sgr A*( ∼ 6 R Sch ) have been observed, which can constraint m − g a γ in a specific window. ⋆ After quantitative estimation, wash-out by resolution and line of sight can be negligible.
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