Listening for Invisible Axions with Gravitational Waves

Ben A. Stefanek

Audible AxionsC. Machado, W. Ratzinger, P. Schwaller, BAS

JHEP 01 (2019) 053 arXiv:1811.01950

The Audible Axion Model (4 parameters)

• Weconsiderthefollowingeffec3vefieldtheoryconsis3ngofanaxion-likepar3cle(ALP)andamasslessdarkphoton

• Weassumesomeexplicitbreakingoftheglobalsymmetryatthescale,whichgeneratesamassfortheALP

�2

S =

Zd

4x

p�g

1

2@µ�@

µ�� V (�)� 1

4Xµ⌫X

µ⌫ � ↵

4f�Xµ⌫

eX

µ⌫

�

<latexit sha1_base64="AQaBXR2yDKoGtrmXTFD4PAe7Hpw=">AAADBHicbVJNj9MwEHXC11K+unCDi0WFtCBaJdVK7AVpBReOi6DdSnVbOc4ktdZxsrHDUlk+cOGvcOEAQlz5Edz4Nzhp1C67jGTp+c17Hs/YUSG40kHwx/OvXL12/cbOzc6t23fu3uvu3h+rvCoZjFgu8nISUQWCSxhprgVMihJoFgk4jk5e1/njD1Aqnsv3elXALKOp5AlnVDtqses9JBGkXBo4rRrqme20jHL1tdtlVC8ZFeadxS8x4VLjeG727UdMnhN1WmrTT63DmEQ8TaeYJCVlJrRmaElBS82pWBiSVU5TLHltatn5ObaPx3s1eNrpb/z7Fk8aJ5FONplvYL+tQagoltQJE9scslWTMx6D5iIGM7FbY3PDGe4QkPGmPUEjEEbQ1K757SAW3V4wCJrAl0HYgh5q42jR/U3inFUZSM0EVWoaBoWembpdJsCdXykoKDuhKUwdlDQDNTPNI1r8xDExTvLSLTfjhj3vMDRTapVFTlm/iLqYq8n/5aaVTg5mhsui0iDZulBSCaxzXP8IHPMSmBYrBygrubsrZkvqBqzdv+m4IYQXW74MxsNBGAzCt8Pe4at2HDvoEXqM9lCIXqBD9AYdoRFi3ifvi/fN++5/9r/6P/yfa6nvtZ4H6J/wf/0Fdif3cg==</latexit><latexit sha1_base64="AQaBXR2yDKoGtrmXTFD4PAe7Hpw=">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</latexit><latexit sha1_base64="AQaBXR2yDKoGtrmXTFD4PAe7Hpw=">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</latexit><latexit sha1_base64="AQaBXR2yDKoGtrmXTFD4PAe7Hpw=">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</latexit>

�<latexit sha1_base64="ZnF0+WI7w4Ygwz1GYRXdsoTIiIM=">AAAB63icbVDLSsNAFL2pr1pfVZe6GCyCq5K40WXRjcsW7APaUCbTSTN0ZhJmJkIJ/QU3LhRx6z/4He7c+SlO0i609cCFwzn3cu89QcKZNq775ZTW1jc2t8rblZ3dvf2D6uFRR8epIrRNYh6rXoA15UzStmGG016iKBYBp91gcpv73QeqNIvlvZkm1Bd4LFnICDa5NEgiNqzW3LpbAK0Sb0FqjdOP1jcANIfVz8EoJqmg0hCOte57bmL8DCvDCKezyiDVNMFkgse0b6nEgmo/K26doXOrjFAYK1vSoEL9PZFhofVUBLZTYBPpZS8X//P6qQmv/YzJJDVUkvmiMOXIxCh/HI2YosTwqSWYKGZvRSTCChNj46nYELzll1dJ57LuuXWvZdO4gTnKcAJncAEeXEED7qAJbSAQwSM8w4sjnCfn1Xmbt5acxcwx/IHz/gMiQpCA</latexit><latexit sha1_base64="ETSvLEBCnQOwl6YUticS5P+8yhQ=">AAAB63icbVC7SgNBFL3rM8ZX1FKRwSBYhV0bLYM2lgmYByRLmJ3MZofMzC4zs0JYUtraWChi6z/kO+z8Bn/C2SSFJh64cDjnXu69J0g408Z1v5yV1bX1jc3CVnF7Z3dvv3Rw2NRxqghtkJjHqh1gTTmTtGGY4bSdKIpFwGkrGN7mfuuBKs1ieW9GCfUFHkgWMoJNLnWTiPVKZbfiToGWiTcn5erJpP79eDqp9Uqf3X5MUkGlIRxr3fHcxPgZVoYRTsfFbqppgskQD2jHUokF1X42vXWMzq3SR2GsbEmDpurviQwLrUcisJ0Cm0gvern4n9dJTXjtZ0wmqaGSzBaFKUcmRvnjqM8UJYaPLMFEMXsrIhFWmBgbT9GG4C2+vEyalxXPrXh1m8YNzFCAYziDC/DgCqpwBzVoAIEInuAFXh3hPDtvzvusdcWZzxzBHzgfPwCHkeY=</latexit><latexit sha1_base64="ETSvLEBCnQOwl6YUticS5P+8yhQ=">AAAB63icbVC7SgNBFL3rM8ZX1FKRwSBYhV0bLYM2lgmYByRLmJ3MZofMzC4zs0JYUtraWChi6z/kO+z8Bn/C2SSFJh64cDjnXu69J0g408Z1v5yV1bX1jc3CVnF7Z3dvv3Rw2NRxqghtkJjHqh1gTTmTtGGY4bSdKIpFwGkrGN7mfuuBKs1ieW9GCfUFHkgWMoJNLnWTiPVKZbfiToGWiTcn5erJpP79eDqp9Uqf3X5MUkGlIRxr3fHcxPgZVoYRTsfFbqppgskQD2jHUokF1X42vXWMzq3SR2GsbEmDpurviQwLrUcisJ0Cm0gvern4n9dJTXjtZ0wmqaGSzBaFKUcmRvnjqM8UJYaPLMFEMXsrIhFWmBgbT9GG4C2+vEyalxXPrXh1m8YNzFCAYziDC/DgCqpwBzVoAIEInuAFXh3hPDtvzvusdcWZzxzBHzgfPwCHkeY=</latexit><latexit sha1_base64="U85VTJ3p878UMP0ysiiXjjYK99o=">AAAB63icbVBNSwMxEJ3Ur1q/qh69BIvgqex60WPRi8cK9gPapWTTbDc0yS5JVihL/4IXD4p49Q9589+YbfegrQ8GHu/NMDMvTAU31vO+UWVjc2t7p7pb29s/ODyqH590TZJpyjo0EYnuh8QwwRXrWG4F66eaERkK1gund4Xfe2La8EQ92lnKAkkmikecEltIwzTmo3rDa3oL4HXil6QBJdqj+tdwnNBMMmWpIMYMfC+1QU605VSweW2YGZYSOiUTNnBUEclMkC9uneMLp4xxlGhXyuKF+nsiJ9KYmQxdpyQ2NqteIf7nDTIb3QQ5V2lmmaLLRVEmsE1w8Tgec82oFTNHCNXc3YppTDSh1sVTcyH4qy+vk+5V0/ea/oPXaN2WcVThDM7hEny4hhbcQxs6QCGGZ3iFNyTRC3pHH8vWCipnTuEP0OcPEkiOOw==</latexit>

Xµ<latexit sha1_base64="iAoE2W8BaH6+lKm18So0vDHw/G4=">AAAB7nicbVC7SgNBFL0bXzG+opY2g1GwCrs2WgZtLCOYByQhzE7uJkNmZpeZWSEs+QgbC0Vs/QJrv8HOD7F38ig08cCFwzn3cu89YSK4sb7/5eVWVtfWN/Kbha3tnd294v5B3cSpZlhjsYh1M6QGBVdYs9wKbCYaqQwFNsLh9cRv3KM2PFZ3dpRgR9K+4hFn1Dqp0exmbZmOu8WSX/anIMskmJNS5eT7/QMAqt3iZ7sXs1SiskxQY1qBn9hORrXlTOC40E4NJpQNaR9bjioq0XSy6bljcuqUHoli7UpZMlV/T2RUGjOSoeuU1A7MojcR//NaqY0uOxlXSWpRsdmiKBXExmTyO+lxjcyKkSOUae5uJWxANWXWJVRwIQSLLy+T+nk58MvBrUvjCmbIwxEcwxkEcAEVuIEq1IDBEB7gCZ69xHv0XrzXWWvOm88cwh94bz9OuJJ4</latexit><latexit sha1_base64="H72AwLZMeBN6j71aqBgYuUUAyTw=">AAAB7nicbVC7SgNBFL3rMyY+opY2g1GwCrs2WgZtLCOYByRLmJ3MJkPmsczMBsKSj7CxUMTWL/AH/AM7P0RrJ49CEw9cOJxzL/feEyWcGev7n97K6tr6xmZuK1/Y3tndK+4f1I1KNaE1orjSzQgbypmkNcssp81EUywiThvR4HriN4ZUG6bknR0lNBS4J1nMCLZOajQ7WVuk406x5Jf9KdAyCeakVDn5ensfFr6rneJHu6tIKqi0hGNjWoGf2DDD2jLC6TjfTg1NMBngHm05KrGgJsym547RqVO6KFbalbRoqv6eyLAwZiQi1ymw7ZtFbyL+57VSG1+GGZNJaqkks0VxypFVaPI76jJNieUjRzDRzN2KSB9rTKxLKO9CCBZfXib183Lgl4Nbl8YVzJCDIziGMwjgAipwA1WoAYEB3MMjPHmJ9+A9ey+z1hVvPnMIf+C9/gBHWJPy</latexit><latexit sha1_base64="H72AwLZMeBN6j71aqBgYuUUAyTw=">AAAB7nicbVC7SgNBFL3rMyY+opY2g1GwCrs2WgZtLCOYByRLmJ3MJkPmsczMBsKSj7CxUMTWL/AH/AM7P0RrJ49CEw9cOJxzL/feEyWcGev7n97K6tr6xmZuK1/Y3tndK+4f1I1KNaE1orjSzQgbypmkNcssp81EUywiThvR4HriN4ZUG6bknR0lNBS4J1nMCLZOajQ7WVuk406x5Jf9KdAyCeakVDn5ensfFr6rneJHu6tIKqi0hGNjWoGf2DDD2jLC6TjfTg1NMBngHm05KrGgJsym547RqVO6KFbalbRoqv6eyLAwZiQi1ymw7ZtFbyL+57VSG1+GGZNJaqkks0VxypFVaPI76jJNieUjRzDRzN2KSB9rTKxLKO9CCBZfXib183Lgl4Nbl8YVzJCDIziGMwjgAipwA1WoAYEB3MMjPHmJ9+A9ey+z1hVvPnMIf+C9/gBHWJPy</latexit><latexit sha1_base64="0D4e2NpvX5M8ONaWCWG0HTFemDw=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5stAzaWEYwH5AcYW8zlyzZ3Tt294Rw5EfYWChi6++x89+4Sa7QxAcDj/dmmJkXpYIb6/vfXmljc2t7p7xb2ds/ODyqHp+0TZJphi2WiER3I2pQcIUty63AbqqRykhgJ5rczf3OE2rDE/VopymGko4Ujzmj1kmd7iDvy2w2qNb8ur8AWSdBQWpQoDmofvWHCcskKssENaYX+KkNc6otZwJnlX5mMKVsQkfYc1RRiSbMF+fOyIVThiROtCtlyUL9PZFTacxURq5TUjs2q95c/M/rZTa+CXOu0syiYstFcSaITcj8dzLkGpkVU0co09zdStiYasqsS6jiQghWX14n7at64NeDB7/WuC3iKMMZnMMlBHANDbiHJrSAwQSe4RXevNR78d69j2VryStmTuEPvM8fhEiPqQ==</latexit>

V (�) = m2f2

1� cos

✓�

f

◆�

<latexit sha1_base64="9WQMN8LS3FxQX1kiD7y8FyJQH40=">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</latexit><latexit sha1_base64="9WQMN8LS3FxQX1kiD7y8FyJQH40=">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</latexit><latexit sha1_base64="9WQMN8LS3FxQX1kiD7y8FyJQH40=">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</latexit><latexit sha1_base64="9WQMN8LS3FxQX1kiD7y8FyJQH40=">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</latexit>

�i = ✓f , �0i ⇡ 0 , ✓ ⇠ O(1)

<latexit sha1_base64="qjZFLiw3lKPEGoNjyG7vIqFBZZc=">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</latexit><latexit sha1_base64="qjZFLiw3lKPEGoNjyG7vIqFBZZc=">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</latexit><latexit sha1_base64="qjZFLiw3lKPEGoNjyG7vIqFBZZc=">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</latexit><latexit sha1_base64="qjZFLiw3lKPEGoNjyG7vIqFBZZc=">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</latexit>

⇤ =p

mf

Vacuum Misalignment

• SincetheALPhasnoreasontobeneartheminimumofthepoten3alwhenit3lts,wegenericallyexpectini3alcondi3onsoftheform

�3

�i = ✓f , �0i ⇡ 0 , ✓ ⇠ O(1)

<latexit sha1_base64="qjZFLiw3lKPEGoNjyG7vIqFBZZc=">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</latexit><latexit sha1_base64="qjZFLiw3lKPEGoNjyG7vIqFBZZc=">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</latexit><latexit sha1_base64="qjZFLiw3lKPEGoNjyG7vIqFBZZc=">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</latexit><latexit sha1_base64="qjZFLiw3lKPEGoNjyG7vIqFBZZc=">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</latexit>

V (�)

��i = ✓f

ALP and Dark Photon Abundances

• TheALPbeginstooscillatewhentheHubbleratedropsbelowtheALPmass

• Forsimplicity,weassumethenoini3alabundanceforthedarkphoton.However,itcanbesourcedastheALProlls.

�4

V (�)

��i = ✓f

⌦osc

� =⇢osc�

⇢osctot

⇡ m2✓2f2/2

3M2

PH2

osc

⇡✓

✓f

MP

◆2

⇢osc� ⇠ 1

2m2�2

iHosc

⇠ m,

Dark Photon Production

• WeassumethattheALPbeginsoscilla3nginapost-infla3onaryradia3on-dominatedFRWuniverse(),inwhichtheALPobeys

andthedarkphotonobeysawaveequa3onsourcedby

�5

✓@2

@⌧2�r2

◆~X = ↵

�0

f~r⇥ ~X , ~r · ~X = 0 .

<latexit sha1_base64="k5LcM/+B/HgUV2skEHsI59kFVug=">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</latexit><latexit sha1_base64="XXbNyufBdwBhFg65J0tkOqfdE4I=">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</latexit><latexit sha1_base64="XXbNyufBdwBhFg65J0tkOqfdE4I=">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</latexit><latexit sha1_base64="qXcdG47eW2/hM3r/dhbH/7Xky9w=">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</latexit>

�00 + 2aH�0 + a2@V

@�=

↵

fa2 ~E · ~B ,

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

�0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

⌦� < ⌦rad

Source for dark photons(while ALP rolls)

Dark E and B, sink for ALP

Dark Photon Production: Quantum Picture

Wequan3zethedarkphotonfieldas

whichleadstothefollowingequa3onforthemodefunc3ons

AstheALProlls,thereexistmomentaforwhichisnega3ve.Thecorrespondingmodesaretachyonicandgrowexponen3ally:

�6

Xi(x, ⌧) =X

�=±

Zd3k

(2⇡)3v�(k, ⌧) "

i�(k) a�(k) e

ik·x+ h.c.<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

v±(k, ⌧) ⇠ e|!±|⌧<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

!2±

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Circular pols. ( )

v00± + !2±(⌧)v± = 0 , !2

±(⌧) = k2 ⌥ k↵

f�0

k⇥ "± = ⌥ik"±

Dark Photon Spectrum and Parity Violation

Becausetheini3alsignofdetermineswhichhelicitybecomestachyonicfirst,thesystemviolatesparityandonehelicityexperiencesexponen3allymoreproduc3onthantheother.

Tachyonicmodes:

Mosttachyonicmode:

�7

2 5 10 20 50

10-25

10-20

10-15

10-10

10-5

1

� / (� ����)

��/�����

�0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

!2+(⌧) = k2 � k

↵

f�0

k(⌧)

0 < k <↵�0

f,

k

m. ↵✓

k =↵�0

2f. ↵✓

2m

Gravitational Waves

Darkphotonmodesintherangewhichwereini3allyinvacuumgrowexponen3allywhentheaxionbeginstooscillate

Theserapidlygrowingmodesamplifyquantumfluctua3onsofthedarkphotonintoa3me-varying,anisotropicclassicalenergydistribu3onwhichsourcesgravita3onalwaves

�8

v±(k, ⌧) ⇠ e|!±|⌧<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

⇧ij(k, ⌧) =⇤klij

a2

Zd3q

(2⇡)3

hEk(q, ⌧)El(k� q, ⌧) + Bk(q, ⌧)Bl(k� q, ⌧)

i.

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

v±(k, ⌧ ⌧ ⌧osc

) =1p2k

e�ik⌧

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

h00ij(k, ⌧) + k2hij(k, ⌧) =

2

M2P

⇧ij(k, ⌧) ,Anisotropic

stressGravity Waves

!2± < 0

0 < k < ✓↵m

The Big Picture

�9

Cold Dark Matter

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

H . m, Particle Production

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

H > m<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

P. Agrawal, G. Marques-Tavares, W. Xue: arXiv:1708.05008

• Theoscilla3ngALPproducesdarkphotonswhichsourceGW.ASerpar3cleproduc3on,theremainingALPabundancecanbeCDM.

Gravitational Wave Spectrum

�10

10-2 0.1 1 10 10010-13

10-12

10-11

10-10

10-9

10-8

10-7

� / (� ����)

����

kpeak ⇡ (↵✓)2/3 m, ⌦GW(kpeak) ⇡✓

f

MP

◆4 ✓✓2

↵

◆ 43

.

Properties of the GW Spectrum: Chirality

TheproducedGWspectrumishighlypolarizedbecausetheGWinherittheparityviola3oninthedarkphotonpopula3on

�11

2 5 10 20 50

10-25

10-20

10-15

10-10

10-5

1

� / (� ����)

��/�����

10-2 0.1 1 10 10010-13

10-12

10-11

10-10

10-9

10-8

10-7

� / (� ����)

����

+ =+ =

�12

10-9 10-7 10-5 10-3 10-1 101 103

10-16

10-14

10-12

10-10

10-8

10-16

10-14

10-12

10-10

10-8

f0 HHzL

h2WGW

0

LISA

BBO

DECIGO

LIGO ETIPTA

SKA

QCD

Axion

1QCD

Axion

2

ALP1

ALP2

ALP3

ALP4

ALP5

ALP6

Listening for Invisible Axions

⌦0GW ⇡ ⌦0

�

✓f

MP

◆4 ✓✓2

↵

◆ 43

f0 ⇡ m

✓T0

T⇤

◆(↵✓)2/3 =

rm

MPT0 (↵✓)

2/3 ,

�13

10-9 10-7 10-5 10-3 10-1 101 103

10-16

10-14

10-12

10-10

10-8

10-16

10-14

10-12

10-10

10-8

f0 HHzL

h2WGW

0

LISA

BBO

DECIGO

LIGO ETIPTA

SKA

QCD

Axion

1QCD

Axion

2

ALP1

ALP2

ALP3

ALP4

ALP5

ALP6

ALPs Detectable by Pulsar Timing Arrays:

f & 1017 GeV

m ⇠ 10�13 - 10�10 eV

�14

10-9 10-7 10-5 10-3 10-1 101 103

10-16

10-14

10-12

10-10

10-8

10-16

10-14

10-12

10-10

10-8

f0 HHzL

h2WGW

0

LISA

BBO

DECIGO

LIGO ETIPTA

SKA

QCD

Axion

1QCD

Axion

2

ALP1

ALP2

ALP3

ALP4

ALP5

ALP6

ALPs Detectable by LISA:

m ⇠ 0.01� 1 eV

f & 1017 GeV

�15

10-9 10-7 10-5 10-3 10-1 101 103

10-16

10-14

10-12

10-10

10-8

10-16

10-14

10-12

10-10

10-8

f0 HHzL

h2WGW

0

LISA

BBO

DECIGO

LIGO ETIPTA

SKA

QCD

Axion

1QCD

Axion

2

ALP1

ALP2

ALP3

ALP4

ALP5

ALP6

Future High Frequency Detectors:

f & 1017 GeV

m ⇠ 102-106 eV

Listening for Invisible Axions• BecauselargedecayconstantscorrespondtolargerGWamplitude,gravita3onal

wavedetectorscanprobeAudibleAxionmodelsfromthe“boVomup”!

�1610-14 10-9 10-4 10 106

10-18

10-16

10-14

10-12

10-10

10-8

10-6

m [eV]

1/f

a[G

eV-

1]

SKA

LISA

BBO

DECIGO ET

Astrophysics

IAXO

ADMX

QCD

Axion

Neff

Cosmological

BH

Superr

adia

nce

Listening for Invisible Axions• BecauselargedecayconstantscorrespondtolargerGWamplitude,gravita3onal

wavedetectorscanprobeAudibleAxionmodelsfromthe“boVomup”!

�1710-14 10-9 10-4 10 106

10-18

10-16

10-14

10-12

10-10

10-8

10-6

m [eV]

1/f

a[G

eV-

1]

SKA

LISA

BBO

DECIGO ET

Astrophysics

IAXO

ADMX

QCD

Axion

Neff

Cosmological

BH

Superr

adia

nce

Thank You!

Backup Slides

Properties of the GW Spectrum: Param Scaling

TheenergyinGWislimitedbytheenergyintheaxionfield

Roughes3mateforthepeakoftheGWspectrum

Es3ma3ngatthe3meofemissionleadsto

�19

⌦osc

� =⇢osc�

⇢osctot

⇡ m2✓2f2/2

3M2

PH2

osc

⇡✓

✓f

MP

◆2

.

kpeak ⇡ k(⌧⇤) , ⌦GW(kpeak) = ⌦2�

✓a⇤H⇤kpeak

◆2

.

⌦GW(kpeak) ⇡✓

f

MP

◆4 ✓✓2

↵

◆ 43

.kpeak ⇡ (↵✓)2/3 m,

Large decay constant requiredALP mass sets peak frequency

1 2 5 10 20 5010-8

10-4

1

104

108

1012

� / ����

ρ ���

(�+��)/ρ ����

ΔNeff = 0.3

ΔNeff = 0.03

ρϕa3

ρXa4

Quick Recap

�20

Cold Dark Matter

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

H . m, Particle Production

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

H > m<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

0.1 1 10 10010-12

10-11

10-10

10-9

10-8

10-7

� / (� ����)

����

a /aosc = 6.7a /aosc = 10.3a /aosc = 66

P. Agrawal, G. Marques-Tavares, W. Xue: arXiv:1708.05008

Energy Density and Gauge Power Spectrum

1 2 5 10 20 5010-8

10-4

1

104

108

1012

� / ����

ρ ���

(�+��)/ρ ����

ΔNeff = 0.3

ΔNeff = 0.03

ρϕa3

ρXa4

1 2 5 10 20 50

10-43

10-33

10-23

10-13

10-3

� � / (� ����)�Ω

�/�����

a /aosc = 6.7a /aosc = 10.3a /aosc = 23a /aosc = 66

More on GW Spectrum

hij(k, ⌧) =2

aM2P

Z ⌧

⌧osc

d⌧ 0a(⌧ 0)G(k, ⌧, ⌧ 0)⇧ij(k, ⌧0)

d⇢GW

d log k=

M2P k

3

8⇡2a2Ph0

(k, ⌧) h0|h0ij(k, ⌧)h

0⇤ij (k

0, ⌧)|0i = (2⇡)3Ph0(k, ⌧)�(k� k0)

d⇢GW

d log k=

k3

4⇡2a4M2P

Z ⌧

⌧osc

d⌧ 0d⌧ 00a(⌧ 0)a(⌧ 00) cos [k(⌧ 0 � ⌧ 00)]⇧2(k, ⌧ 0, ⌧ 00)

h0|⇧ij(k, ⌧)⇧⇤ij(k

0, ⌧ 0)|0i = (2⇡)3⇧2(k, ⌧, ⌧ 0)�(k� k0)