Gravitational wave forest from axions

Naoya Kitajima
FRIS, Tohoku U.
Collaboration with Y. Urakawa, J. Soda,
P. Agrawal, M. Reece, T. Sekiguchi, F. Takahashi
Axion Cosmology 2019/5/14 in YITP

Axiverse
Arvanitaki et al. (2010)

String theory predicts many axions with broad mass range

Constraints on axion/ALPs(see also Obata, Ito, Kato, Chiueh, Hayashi’s talks)Credit : Particle data group

String axions with large decay constant ~1016 GeV or small gaγγ—> it is diﬃcult to search them by direct detection experiments

String axion search via Gravitational wave (GW)
GW can be sourced by scalar field fluctuations

GW signal becomes strong for larger decay constants ⇠ 1016 GeV<latexit sha1_base64="Ff5NH8xzppin3jTGV7UyDD7MYqM=">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</latexit>Future GW observation with multi-frequency bands

Moore et al. 1408.0740

Axion mass (m) <—> frequency

Contents
1. GW production from axion self-resonance• Flapping resonance• Oscillon formation• GW forest
2. Tachyonic gauge field production from axion• Dark photon dark matter production• Helical GW production
Axion as DM in radiation dominated Universe

GW production from axion self-resonance
NK, J. Soda, Y. Urakawa, arXiv:1807.07037

Axion potential
Cosine-type potential :(Dilute instanton gas approximation)(some classes of ) string axion
Potential with plateau region
e.g. pure-natural potential
Nomura, Watari, Yamazaki (2017)

String axion with α-attractor type potential
m2 f 2 tanh2 ( /f )V( )=2 1 + c tanh2 ( /f )Soda, Urakawa, 1710.00305- Delay of the oscillation onset- Tachyonic instability (at V’’ < 0)strong instability

Evolution equations (Minkowski background)Zero-mode :<latexit sha1_base64="zjZbH+O8amtU3lc9tfIHsANbSB8=">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</latexit>

Fluctuation (nonzero mode) :<latexit sha1_base64="8usNFQ8NWLZfuu4jv66INBmhsQE=">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</latexit>¨ + dV = 0d✓◆2¨ + k2 + d Vkd 2tachyonic

potential
2 2

2

m f
tanh ( /f )V( )=2 1 + c tanh2 ( /f )
✓
◆
2
1
d V
2
2
Vk ( k ) =
k +
k
2
d 2
<latexit sha1_base64="NAPOJ+OHcEEB0vWEPeIJwqz40M4=">AAAC6XichVFNa9RQFD1N/ait2mndCG5Ch8oUobwEoaWgFLpx2a+ZFprOkGReZh55k4TkzUAN+QPuXLXUlYoLce8fcOPCbRf9CcWdLbiwC28yEdHBekPyzjvvnpPzuE4kRaIYOx3Txq9dv3Fz4tbk1O07d6crM7ONJOzHLq+7oQzjXcdOuBQBryuhJN+NYm73HMl3HH8tP98Z8DgRYbCtDiK+37M7gfCEayuiWhWv0Uotx9P9rGa1uVS2FXXFL2pBf6JbXmy7qZGlZmZJ7qma3zQfDcl202xkaTtXNM1Mt2LR6aoFfdSnabYqVbbIitJHgVGCKspaDysfYaGNEC766IEjgCIsYSOhZw8GGCLi9pESFxMSxTlHhknS9qmLU4dNrE/fDu32Sjagfe6ZFGqX/iLpjUmpY56dsPfsnH1mH9gZu/ynV1p45FkOaHWGWh61pl/c3/r+X1WPVoXub9WVmRU8LBdZBWWPCia/hTvUD54fnm+tbM6nD9kb9pXyv2an7BPdIBhcuO82+OarK/I4lCWj8Rh/D2MUNMxFg/DG4+rq03JQE3iAOdRoGktYxTOso07uX/ANP3Cp+dpL7Ug7HrZqY6XmHv4o7e1PHhu4kw==</latexit>

k =0Flapping resonancefluctuation<latexit sha1_base64="e3VAcgI5QGW9Nk2yc7U2TyA1G6E=">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</latexit>

Vk (<latexit sha1_base64="H/hAElyj5z3zAx4yIWFEkTOzVeo=">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</latexit>

V( )

k)

homogeneous mode/<latexit sha1_base64="gFu/k2M51HC/5n2D5jDMg2//SPw=">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</latexit><latexit sha1_base64="bQES5Nrh8696g7LooubHjrJDMSA=">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</latexit>i

Fukunaga, NK, Urakawa 1903.02119 —> Fukunaga’s talk

Lattice simulation
O(1) overdensity

oscillon(see also next Takashi’s talk for oscillon formation after inflation)

Lattice simulation

1283 grids

Axion evolution and spectrum
delay

spectrum

101

~⟩
⟨φ
~2⟩)1/2
(⟨δφ
~⟩)
~
ρ(⟨φ
~)
~(φ
δρ

0

10

10-1

nk [mf2]

10-2
10-3
10-4
10-5
-6

10

-7

10

0

10

20

30

40

100
10-2
10-4
10-6
10-8
10-10
10-12
10-14
10-16

time evolution50

1

c = 2,

i = 3fmτ
<latexit sha1_base64="VpehoM2V3vW6ocXeto7hidhXolc=">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</latexit>

10
k/m

100

Gravitational wave emission
ds =dt2 + a2 (t)( ij + hij )dxi dxj

tensor metric perturbation (GW)
evolution equation (Einstein equation) for gravitational wavesḧij + 3H ḣij
<latexit sha1_base64="/idgn40FCIseCgCKjwgZqhvzeUw=">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</latexit>

r2 hij
TT
=
16⇡G⇧
ij
a2

Density spectrum of GW

with

⇧TT
ij =
<latexit sha1_base64="3slRqf6dxL8V1z2K+eU1NqmV8tk=">AAADRXichVJNaxRBEK2MX/lQs+pF8DK4rHhx6QmCIigBL55kk+wmkUx26Bl7k872fDDTu7IO+wc8evHgScGDePUfeIh/wEO86iUEQciiF0Hf9I4EDcYeprtedb3XXdXlJ0pmmrGdCevY8RMnT01OTc+cPnN2tnLu/HIW99JAtIJYxemqzzOhZCRaWmolVpNU8NBXYsXv3i32V/oizWQcNfUgEesh34hkRwZcw+VV7rsN2c7dNLSbzaGXy62hfdu+5nZSHuTOMOftuaHdaOcqLDfdhKdacuVJmJvyAG8Z7FWqrM7MsA8bTmlUqRyNuPKVXHpIMQXUo5AERaRhK+KU4Vsjhxgl8K1TDl8KS5p9QUOaBreHKIEIDm8X8wbQWumNgAvNzLADnKLwp2DaVGMf2Gu2z96zN2yX/finVm40irsMsPpjrki82ScXl77/lxVi1bR5wDqC4SP66Jw0deimyUUit8R4iiyDsX7/8bP9pVuLtfwKe8n2kN8LtsPeIcOoPwpeLYjF51CvQT81LEGPTE1Cc8sIr+BCv7i3RJ0GwMLgHpDGmQny6Je12zbV24P27yo+pXuIV4gQ5Rtx1n0w1RmhJZy/G+CwsTxXd1jdWbhenb9TNsckXaLLdBUdcIPmod6gFlTf0kf6RJ+tbeuLNbK+jUOtiZJzgf4Y1s9f3VDPaQ==</latexit>

1 lm
Pij @i @j
2a

TT projection tensor

1 d⇢GW
1⌦GW (k) =, ⇢GW =
hḣij ḣij i⇢cr d ln k
32⇡G

Evolution of GW spectrum
10-6

-10

10-15

ΩGW

ΩGW

10

10-20

10-8

10-25
10-30

10-7

1

10

10-9

100

1

10

k/m

k/m

c = 2,<latexit sha1_base64="0oXzjYG4YY6lgJBad3KRITHfAWA=">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</latexit>

GW frequency : f0 = 0.7 nHz<latexit sha1_base64="Yugis9/jtJNIQqnUc4ZxCshiofE=">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</latexit>

f = 1016 GeV

i = 3fk ⇣m<latexit sha1_base64="uZTDyp7/dGDFHq+ow6og+pWg5lE=">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</latexit>m
10 12 eV

⌘1/2 ✓ m ◆1/2
Hem

100

Gravitational Wave forestc = 5 and-6

i = 3 (upper),<latexit sha1_base64="2KHdN2FbDeAOcd3S1+1FDo+jU88=">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</latexit>

2 (lower) f = 1016 GeV<latexit sha1_base64="uZTDyp7/dGDFHq+ow6og+pWg5lE=">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</latexit>

10

SKA

LISA DECIGO

10 15 eV

1 µeV

-8

ET

10
ΩGWh

2

-10

10

-12

10

-14

10

-16

10

<latexit sha1_base64="+aZ3/jw5XZqMQjd7gHMOOesE9ag=">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</latexit><latexit sha1_base64="rEcZ04ZH2JMyq8mJVGrNKD4wgGc=">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</latexit>

1 eV<latexit sha1_base64="AJWMRjGYsM4MTqmW8KattYs2bLI=">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</latexit>

1 MeV<latexit sha1_base64="bJHNhWjmWkPDm/vZZbaFSqurF4E=">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</latexit>

-18

10

10-10 10-8 10-6 10-4 10-2 100 102 104
f [Hz] NK, Soda, Urakawa,1807.07037! axions are easily overproduced

Axion = dark matter-6

c = 5 and<latexit sha1_base64="KUJ8b5v8zUxJPuWTk25PqxZsALY=">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</latexit>i = 3 (left),

2 (right)

10

-8

SKA

10

1016 GeV

-10

ΩGWh2

10

<latexit sha1_base64="c0HmcSCwfZnAi9pfwsHRf+EMelw=">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</latexit>

-12

10

1015 GeV-1410<latexit sha1_base64="UAKgsMuNuum86yfYkJTpgQk6JKw=">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</latexit>

-16

10

-18

10

10-20 -15 -14 -13 -12 -11 -10
10 10 10 10 10 10
10-9 10-8
f [Hz]
2⌦ h = 1.5<latexit sha1_base64="KaMaKMoeoomzjIvlYLgKmCtcyVU=">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</latexit>⇣m
10 14 eV

⌘1/2 ✓ m ◆3/2 ✓
Hosc

osc
1016 GeV◆2

Tachyonic U(1) gauge field production from axion
Agrawal, NK, Reece, Sekiguchi, Takahashi, 1810.07188
NK, J. Soda, Y. Urakawa, in prep.
Non-Abelian case —> Peter’s talkmagnetogenesis —> Kamada & Teerthal’s talk1. Dark photon DM production from axion
Agrawal, NK, Reece, Sekiguchi, Takahashi, 1810.07188<latexit sha1_base64="+6jvEORCDunc8EpX/zEeWjqG2Wo=">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</latexit>

1 µ
L = @ @µ2

V( )1Fµ⌫ F µ⌫4

1 2
m 0 Aµ Aµ24fa

Fµ⌫ F̃ µ⌫dark photon mass

Equations of motion<latexit sha1_base64="zYuCObumCUcrm1vNWqhN/pZQ5C0=">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</latexit>with: Lorentz gauge condition<latexit sha1_base64="fx2adkiFVr/RxPrirZ6DpFZd3mk=">AAACpXichVFNSxtRFD1OtfWrNdZNwc2QoE0XlZsiWAqKxY0bQaOJgcQOM9Nn+nC+nHkJaMhe/AMuXCl0UQSX2n03/gEX/gRxqeDGhXcmA6WVpnd488497577zuNagSMjRXTVoz3r7Xv+on9gcGj45auRzOjrcuQ3QluUbN/xw4plRsKRnigpqRxRCUJhupYj1q2thfh8vSnCSPremtoJxIZr1j25KW1TMWVksrXADJU0HaPmNvK1aDtUrff1tv75C+f6O31WJyOToylKQn8KCinIIY1lP3OGGr7Ch40GXAh4UIwdmIj4q6IAQsDcBlrMhYxkci7QxiBrG1wluMJkdov/dc6qKetxHveMErXNtzi8QlbqmKBL+kG3dEEndE0P/+zVSnrEXnZ4tzpaERgj+29W7/+rcnlX+PZb1dWzwiY+Jl4lew8SJn6F3dE3dw9uVz8VJ1qTdEw37P+IrugXv8Br3tnfV0TxsIsfi720eTyFv4fxFJQ/TBUYr0zn5ufSQfVjHFnkeRozmMcillHi7ns4xTl+am+1JW1NK3dKtZ5UM4Y/QjMeAc4hnIw=</latexit>

Decomposing gauge field into circular polarization modes :<latexit sha1_base64="8kS3VN2s5VIb1WvqXhbHoIa5/RA=">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</latexit>tachyonic
Initially misaligned axion can produce
dark photons via tachyonic instabilityV( )<latexit sha1_base64="k6VYGX5IvmuIly7Tz1AV0ywKZpI=">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</latexit>

γ’<latexit sha1_base64="WFuiiBHoEQit7c/cWGGeRPFT5GQ=">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</latexit>

New production mechanism of dark photon DM
See also, Co, Pierce, Zhang, Zhao, 1810.07196
Bastero-Gil, Santiago, Ubaldi, Vega-Morales, 1810.07208

Numerical results : time evolution(comoving number density)<latexit sha1_base64="NoE07/3QYGcvmvvYgVR2hJlmJhY=">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</latexit>

axion

0

10
na,γ’/na(β=0)= 40, fa = 1014 GeV, ma = 10 8 eV, m 0 = 0.1ma

dark photon-110

tachyonic
growth-210mγ’=0.1ma
0.4ma

-3

10

(cosine potential)

1

10
maτ

axion abundance is suppressed & dark photon becomes dominantNumerical results : power spectrum(comoving number density)
3

10

2

10

<latexit sha1_base64="NoE07/3QYGcvmvvYgVR2hJlmJhY=">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</latexit>= 40, fa = 1014 GeV, ma = 10 8 eV, m 0 = 0.1ma

peak

1

10

increase

0

10

-1

10

-2

10

decrease

-3

10

10-4
-5

10

gauge
axion

10

100

axion massk/mDark photon spectrum has a peak at low momentum(determined by the coupling)Numerical results : relic abundance
Agrawal, NK, Reece, Sekiguchi, Takahashi, 1810.07188

0Ωa,γ’h2

10axion w/o dark photon production

da

-1

n
o
t
o
rk p h

10

-2

axion

10

-3

10

<latexit sha1_base64="W8RRguKsfAdFa7ps71AyAaTo6nY=">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</latexit>

0

= 40

0.1

0.2

0.3
0.4
mγ’/ma0.5

0.6

Dark photon can be the dominant component of DM

2. GW emission with circular polarization
NK, J. Soda, Y. Urakawa, in prep.

ḧij + 3H ḣij
<latexit sha1_base64="wjsTgHrtkgqzdGqhOfq/NA8HmTw=">AAAC4nichVHLShxBFL22MVFj4qgbwU2RQRHCDNWjqAiC6EKXo86oYDtNdU+NU7FfdNcMaNE/4E5cuEg2CWQRss8PhIA/4MJPCC41ZOPC2w/IQ2Ju012nTt1z+hTXChwRSUqverTeJ31Pn/UPDD4fevFyuDAyuh35ndDmddt3/HDXYhF3hMfrUkiH7wYhZ67l8B3rcDU53+nyMBK+V5NHAd932YEnWsJmEimzYBrNpi9VOzaVeBOT12Rm3fiNKBmtkNlKjxVrVGJieMxyGGlnzUukpM8RIxBkjbCGKiUNVdFQRuiSWi0zMAtFWqZpkYdAz0ER8qr6hS9gQBN8sKEDLnDwQCJ2gEGEzx7oQCFAbh8UciEikZ5ziGEQtR3s4tjBkD3E7wHu9nLWw33iGaVqG//i4BuiksAkvaSf6A29oJ/pd3r3Ty+VeiRZjnC1Mi0PzOGT8a2f/1W5uEpo/1I9mllCCxbSrAKzBymT3MLO9N3j85utxc1JNUU/0GvM/55e0a94A697a3/c4JtvH8ljYZZkPPrfw3gItitlfaZc2ZgtLq/kg+qHCXgF0ziNeViGdahCHd2/wTXcwg+tqZ1op9pZ1qr15Jox+KO0d/cgNrUF</latexit>

⇧ij =

@i @j

<latexit sha1_base64="DiIahYmJ+jwiZt7cnrKFDOc1Ijg=">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</latexit>

1
rh
=
ij
a2

1
+ Ei Ej + 2 B i B ja16⇡Ga 2 ⇧TTijelectric & magnetic field
contributions

circular polarization of GW+
hij (k) = h+ek ij (k) + hk eij (k) (Fourier mode)<latexit sha1_base64="iz6yi1vIGv+tYohJdv8x/sMou/A=">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</latexit>

±
±
±
e
(k)
=
e
(k)e
Circular polarization tensor : ijij (k)<latexit sha1_base64="oq4uJwB0ydXNo6TtqFm3/2kWGQc=">AAACwHichZFNL8RQFIZf9f092EhsGhPCZnIHCZEQYWNphkFimLR1Z1zTL21nEpr+AX/AwgqxEHt/wIYtsfATxJLExsKZThEEp2nvuc897+l7c1RbF67H2H2NVFtX39DY1NzS2tbe0Rnr6l52rZKj8Yxm6Zazqiou14XJM57wdL5qO1wxVJ2vqMW5yvlKmTuusMwlb9fm64ZSMEVeaIpHKBeb4ht+1jaCnC+2gyE/q+blYjAsT8nvXHzCd7T9gXKxOEuwMOSfSTJK4ohiwYpdIItNWNBQggEOEx7lOhS49KwhCQab2Dp8Yg5lIjznCNBC2hJVcapQiBbpW6DdWkRN2ld6uqFao7/o9DqklDHA7tgZe2JX7Jw9sNdfe/lhj4qXXVrVqpbbuc793sWXf1UGrR62PlV/evaQx0ToVZB3OySVW2hVfXnv4GlxMj3gD7Jj9kj+j9g9u6QbmOVn7TTF04d/+FHJS0DjSX4fxs9keSSRHE2MpMbiM9PRoJrQh34M0TTGMYN5LCBD3U9wjRvcSrPSlmRJO9VSqSbS9OBLSHtvq1eoBw==</latexit>

Circular polarization vectork ⇥ e± (k) = ⌥ie± (k)<latexit sha1_base64="Qs8IhHQBIrcoYDNRqPFiBYwS7fQ=">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</latexit>

Numerical results : Time evolution of gauge/axion(circular polarization modes)
010Preliminary

-1

10

-2

2 2

ρ [m f ]

10

-3

10

-4

10

-5

10

+ mode− modeδρφρφ
ρφ (no gauge)

10-6
-7

10

-8

10

5

10 15 20 25 30 35 40 45 50mτtanh-type axion potential, c=0 & β=5& dark photon is masslessNumerical results : GW spectrum(circular polarization modes)
NK, Soda, Urakawa in prep
ΩGW (in RD)

10-6+ mode− mode

10-7

-8

10

Preliminary
-9

10

10

100
k/m

Circular polarization asymmetry
see also Adshead+ 1805.04550, Machado+ 1811.01950, Peter’s talk

Summary- String axion whose potential has a plateau region (e.g. alphaattractor model) can show resonance instabilities, leading to GW
emission.

—> Multi-band GW observation can be a powerfullprobe of string axions with wide mass range- Axion-photon interaction in the early Universe can produce
correct relic abundance of dark photon dark matter andsuppress axion abundance—> It can predict circular polarization of GW

Lattice simulation
r2@V
m2 f 2 tanh2 ( /f )+= 0 with V ( ) =2a@2 1 + c tanh2 ( /f )¨ + 3H ˙'

00<latexit sha1_base64="RJWApKB2DDboGu3VQ+dsBBFwn+k=">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</latexit>

2

r '

a00
3 @V
'+a
= 0 with dt = ad⌧,a@<latexit sha1_base64="U6yJWt0XBpNvB3G23/1VCK1Vf6Q=">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</latexit>'=a

RD flat-FRW universe : a / ⌧, a00 = 0<latexit sha1_base64="Um4o0rTeeHM/vgncwMUG97joY9k=">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</latexit>(t, x) ! (n, i, j, k)(t, x, y, z) =

✓

iL
jL
kL
t0 + n t,,,
Ngrid Ngrid Ngrid◆(i, j, k)(i + 1, j, k)EoM in flat-FRW spacetime & temporal gauge(ds2 = dt2

a2 ij dxi dxj & A0 = 0)<latexit sha1_base64="3TXtFpx4lYb2/OCvRB7dq+ypazk=">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</latexit>— Equation of motion for axion —¨ + 3H ˙<latexit sha1_base64="C7XXRSXs4QVmwtE4h0wdA6vu+EM=">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</latexit>

1 2
@V
↵
µ⌫
r
+
+
F
F̃
=0
µ⌫
2
a
@
4f
H = ȧ/a ••• Hubble parameter<latexit sha1_base64="st7SmJNfP4e54X8F9BCvCNF4vZk=">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</latexit>— Equation of motion for gauge field —Äi + H Ȧi<latexit sha1_base64="L7RbVsA14zC3xYMHnjpTXroerjk=">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</latexit>@i Ȧi<latexit sha1_base64="6xmg7ejGo0uclCgtTgwyw0zM648=">AAAC3XichVG/axRBFP6y8UeMP3KJjZBm8YjYeLyLgkFQEmws88NLgtmwzG7mcpOb2x125w7isqWNiG0KK4UUYm8t2PgPWOQfCAQLhQTSWPhubyFqML5ldr73zfvefMMLjFapJdobcobPnb9wceTS6OUrV6+NVcYnltO4m4SyEcY6TlYDkUqtItmwymq5ahIpOoGWK0H7cf98pSeTVMXRU7tt5HpHbEaqqUJhmfIrzzwjEquE9pXrbcQ2m8sZ3XG9ZiLCzBPatESeNUXuetKkSrMmU1ttTn/TmZY6ybfcOb/tPnTJr1SpRkW4p0G9BFWUMR9XPsLDBmKE6KIDiQiWsYZAyt8a6iAY5taRMZcwUsW5RI5R1na5SnKFYLbN/03O1ko24rzfMy3UId+ieSWsdDFFX+k9HdIX+kAH9POfvbKiR9/LNu/BQCuNP/byxtLxf1Ud3i1aJ6ozPVs0MVN4VezdFEz/FeFA33u+c7j0YHEqu0Xv6Bv7f0t79JlfEPWOwt0FufjmDD8Be8l5PPW/h3EaLE/X6ndr0wv3qrOPykGNYBI3cZuncR+zeIJ5NLj7J+zjO344vvPCeeW8HpQ6Q6XmOv4IZ+cXFv60Yw==</latexit>

1 2
1
r Ai + 2 @ i @ j Aj2aa↵
✏ijk ( ˙ @j Ak
fa@j Ȧk ) = 0↵✏ijk @i @j Ak = 0 (modified Gauss’s law)fa
! longitudinal mode can be excited