Gravitational waves
from gravitational
Chern-Simons term
Atsushi NARUKO[Center for Gravitational Physics, YITP]
in collaboration with
R. Namba [McGill] & K. Yonekura [Tohoku]

Relavant action✓ In the first day, Soda-san has introduced
the “relevant action” in this field/conference

L=R

(r )

2

V( )

<latexit sha1_base64="JWQ7+IsPhjGxiGA7CIZ/rId+QGY=">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</latexit>FF⇤<latexit sha1_base64="LcmPIsPETXLzZ8IZ3/ixDiaRocg=">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</latexit><latexit sha1_base64="StPaP33wrHOI5q1khitcV8LlVfU=">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</latexit>

this talk

RR⇤

Contents✓ introduction & motivation✓ model✓ analysis & (preliminary) result✓ conclusion & discussion

primordial

Gravitational waves
from gravitational
Chern-Simons term
Atsushi NARUKO[Center for Gravitational Physics, YITP]
in collaboration with
R. Namba [McGill] & K. Yonekura [Tohoku]

inflation✓ inflation is the accelerated expansion…

- see Peter’s talk- see Peter’s talkpredictions of inflation✓ due to the quasi-exponential expansion of the universe..
→ quantum fluctuations generate tensor perturbations = GWs
whose amplitude is O(H) and hence nearly scale-invariant

hij ⇠ O(H)<latexit sha1_base64="zlTyRPUzxSxYhnlrmiFxWeFQnf0=">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</latexit>✓ the Friedmann equation relates H with ρ :2
2
3Mpl H = ⇢<latexit sha1_base64="Y+M+CLTm7OBB5dtWfvftu9q5Q+I=">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</latexit>☞ observing hij ~ H will determine the energy scale of inflation ??

NO !!!

h>H

see also- Peter’s talk & work
- Namba, Peloso, Shiraishi, Sorbo, Unal [2016]- Dimastrogiovanni, Fasiello, Fujita [2017] …

Spectator axion-SU(2)+<latexit sha1_base64="OhU0cq2gl92kRgO1uu8rf5/CuDk=">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</latexit>

4f

a
Fµ⌫
F̃ aµ⌫

sourceahij = O(H) + hij[A ]<latexit sha1_base64="4gzNrE99ckBBiTHskz0MVmVsrj4=">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</latexit>> O(H)
1, VEV at BGaĀ =<latexit sha1_base64="UNK6+EqgABHL0+vyLwM/pnuDRr8=">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</latexit>(t)

2, chirality in SU(2)
a
i

AR

AL
<latexit sha1_base64="m55fyKd8TDZWk6UCFTqZJGD0rVM=">AAAJ13iclZZLb9NAEMenpUAdXi0cuVhESJyipFSCY5v0JTVSSyFpUakiPzaNVSexbOelKOKGuHDgAp+AD8W34b9jJySsGi+2kp2dnd/szHo1u3bge1FcLP5eWb2zdvfe/XUj9+Dho8dPNjaf1qNuL3REzen63fDCtiLhex1Ri73YFxdBKKy27Ytz+6Yix8/7Ioy8budDPArEVdu67nhNz7FiqN7vNqqNjXyxUOTHVIVSKuQpfU67m2sN+kQudcmhHrVJUIdiyD5ZFOG9pBIVKYDuisbQhZA8Hhc0oRzYHqwELCxob/B/jd5lqu2gL31GTDuYxccvBGnSy9TGhdxkbdLK+c0529vmGLNvGeMIrZ36bEMbUwvaLG5qqcvJnGJE+JZz8RBnwBqZpbOQUROtj36M+OX/CJYCkgsqhORA50ObaOQcIdpkXWXmLV5ni+0EpGW52Ig8mdvmFXNnmeT4/YXoBA14Pds81oHNGHoHsk+7oMdzfZM1Ew2yrJBlTbKikBVNck8h9zTJfYXc1yQPFPJAkzxUyENN8kghjzTJY4U81iSrClnVJE8U8kSTPFPIM02yppA1TbKukPVM0sU7pUKMmNxfztg0nDE2M8NMRtZLm6mpLOvwciZgi6mU1GeZVxYVzFENbmWtCTS420iT5RbmX+5DzDxIKYK9z3V/OVVli/n92shk+rN4kp7FcWZH2J+LMaH+J9I2fxF5cl6l339ak+XeGVMeI5PMvZCcaYteZI0fansYYM5FDwNk4MI65kxcPkH0fIV8Ui96k6fTkG8L07N6PLPT9St3eI/vElme/1ou+MaNp/Tv/UYV6luF0uvC1rvt/E45vfus03N6Qa/g6Q3toL6eooI4OFW/0Q/6aXw0PhtfjK+J6epKyjyjhcf4/ge0CQy8</latexit>

3, (sourced) chiral GWs
sourceahij[A ]<latexit sha1_base64="NBhPbYXrToXvc/zFTRoukkEwor0=">AAAKIHiclZZLb9NAEMenpUBTXi0cuURElThVSUGCY0n6klqppSJppRIiPzatqWNbttMkivJlkHLnY3BDXJDgAh+F/47tkLBqvMSKd3Z2fvNYr3bXDFwnisvlHwuLt5Zu37m7XFi5d//Bw0era48bkd8NLVG3fNcPz0wjEq7jiXrsxK44C0JhdExXnJpXNTl+ei3CyPG9d/EgEM2OceE5bccyYqj81XUaU0iCbBrSJbXwdqhIH2lEHzDSIQOaiHzqwsqC3Qij59CNyaQ25DdsaVCTRq3VUnmjzL+iKlRSoUTp79hfW2rRe0T24bmLWII8iiG78BfhOacKlSmAromIBjKIkZ0cl3msgO3CSsDCgPYK7wv0zlOth770GTFtIYqLf8gVrKc2NuQ2a5NWxi9O2d4UY8i+ZY4DtGbqU85WjFmMcrnMUpeTNcXI8DXX4iDPgDWySmumojZaF/0Y+cv3AJYCkg1KfmcLOhfaRCNjhGiTeZWVX/I8G2wnIM2rxUTmSWyTZ8yeVLLCzxjZCerxfHZ4zON1NobGgL1cO8OpfrKaRhpkVSGrmmRNIWua5LZCbmuSOwq5o0nuKuSuJrmnkHua5L5C7muSBwp5oEkeKuShJnmkkEea5IlCnmiSdYWsa5INhWzkkjaejAoxUuT+fMak/oQxmennMnK/NJnKZLkPz2cCtsikZH+WdeVRwRTV4lbuNYEGdxNZZPkS8ef7EBMPUopg7/K+P586ZIvp9drKZa4n+SQ9g/PMz/B6KseE+p9MO/xF5MnZTL9/tifLtTOkEkZGuWshOdNmvcg9vq/toYeYsx56qMCGdcyV2HyC6PlK7iez3uTp1OfbQnZWDyd2un7lCu/yXSLP81/LGd+48VT+vd+oQmNzo/JiY/Pty9JWNb37LNNTekbP4ekVbWF/PcYOYtFn+k6/6HfhU+FL4WvhW2K6uJAyT2jmV/j5B25ZHfM=</latexit><latexit sha1_base64="GRKofK/kt3l7pgn3Q5Q171zQSh4=">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</latexit>

isotropy of BG

axion-SU(2) term

GW source at 1st ordermore simple model ??
✓ OK.. the result is great..
but we have to assume

I don’t want to speak ill of
the previous models at all…

1, SU(2) fields have to couple with axion [parity-violation]
2, SU(2) fields have specific BG VEVs
3, and then SU(2) fields source GWs+<latexit sha1_base64="+SDwrwOrXxon4sWOe92exDwbzgk=">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</latexit>

⇤

R R̃

parity-violationin gravity

chiral GWs
w/o SU(2)The model✓ model : GR + GCS + axion (spectator) + inflation
2
Mpl

1
2
L=
R+
R R̃ + P [(r ) ] + inflaton
2
16 ⇤<latexit sha1_base64="CNKIzHUe9aHzwApvvPYabWKgcJ8=">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</latexit>

R R̃ = ✏↵<latexit sha1_base64="S/krXa8/w36I8AXX/HlVjk4nqQw=">AAAKsHiclZZdT9swFIZPGdtY9gXb5W6iVZN2MaEWJm03k7ryKYEEq9aCxFiVpG7JyJeSFIqq/hx+1P7NXp+kpcGi8RIpPj4+z+tjx7JtR56bpLXa38rSo+XHT56uPDOev3j56vXq2ptOEg5jR7Sd0AvjU9tKhOcGop26qSdOo1hYvu2JE/tyS7afXIk4ccPgZ3oTiXPfGgRu33WsFK5w1acWmXRLn/ibkkse9UjQGP4JfN/YLyiihNtCCug3Wm/JQi2iC5QywkZMmtsDlD7erCbVPG6Tei3qLqAnaJvk+j4NuS1AOU8+pG7eZ7qr1dp6jR9TNeq5UaX8OQ7Xlrv0C4ohOeB9KAdQdqBvYfQJnVGdasg6pXNkYlHM8yXbBTI0wA4RJefKgvcS3wFqZ7k34LwFbEk76EXOZsxj+5DH9GD32ZuVsn9zLvahPsasLXO8QWnnmnKOUsxxUspNI3U5OaYUGX7lsbjIM2KPHKVTGFEfpYd6ivzl9waRAlYPVAzLgc+DN/PIPmKU2bzKkV/wPFscJ2AtGouNzLO+bZ6x3mwkBr9yXQi65vnMVlCAGLmmHF6P33n93dVN9kw0yKZCNjXJLYXc0iS3FXJbk9xRyB1NclchdzXJPYXc0yT3FXJfkzxQyANN8lAhDzXJI4U80iRbCtnSJNsK2dYkOwrZKSV7eKdUjBaT64sZm0YzxmZmVMrI/dJmamp7fEIsYqL8DMmsbH+W4yqjojmqy6XcayIN7iHSZPsC/S/WEDOF4hm7mDrkiPn12i1lrmb5ZDWL8yzP8Goux4z6n0x9/iPy5DzP//90T5ZrZ0xVtExK10J2phVV5B4/0la4Rp9FhWuMoMf3jrs7j55WzCd1UU2eTiO+LUzP6vEsTldXrvAh3yXKlO8iC9q48dTv329Uo7OxXt9c3/jxudpo5nefFXpH7+kjlL5QA/vrMXYQp9Ko9CthJTI2jFOja1hZ6FIlZ95S4TH+/AMRmTdt</latexit>R↵ µ⌫ Rµ⌫generic term
w/ shift-symmetry

Sato, Kanno & Soda
✓ Soda-san is great

RGB aﬀects BG universe & GWs

The model✓ model : GR + GCS + axion (spectator) + inflation
2
Mpl

1
2
L=
R+
R R̃ + P [(r ) ] + inflaton
2
16 ⇤<latexit sha1_base64="CNKIzHUe9aHzwApvvPYabWKgcJ8=">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</latexit>

R R̃ = ✏↵<latexit sha1_base64="S/krXa8/w36I8AXX/HlVjk4nqQw=">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</latexit>R↵ µ⌫ Rµ⌫generic term
w/ shift-symmetry✓ BG evolution :2
2
3Mpl H =<latexit sha1_base64="FZGOMyN9gvTm15wfgfg9lvCMdcQ=">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</latexit>ρinf. + ρaxion

GWs from GCS✓ GWs propagating in z-direction
0<latexit sha1_base64="zUbOtsAe46qOJ9QfKmiNyWhY34E=">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</latexit>h+2@
gij = a h⇥
0

h⇥
h
0

1
0
0A
0

✓ quadratic action for GWs :SGW =<latexit sha1_base64="vAEenAB6RITF18BhtEAdCy8y2HI=">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</latexit>

1
LGR =
2
<latexit sha1_base64="FzBoswZ8cvUUvFEvoYHTyOuynKM=">AAAKfHiclZZ/T9NAGMefISpUVNA//WdxmmgMZJtGTdQE+Z2wBEQ3SGAs/XFjDd3atAVGKi+HF+WbMX7vaTc2Lqxnm7bPPfd8nnue6+W5swLPjeJy+U9h6t70/QcPZ2aNR3OPnzydX3jWiPyz0BZ12/f88MAyI+G5PVGP3dgTB0EozK7liX3rdFX275+LMHL93q/4MhDNrnnSc9uubcZQ+fM/6ZpsMsmjGrUooU3aoysq0jc819SmEH029BVoE6ribUDvkUBfTG9g9Rtth3y0Euqgv4V2QF3oj2Fv0OItT6esl97MTJI+Orc4yYTk0gl6Ynrbmi+Vl8p8FVWhkgklyq5df2G6RUccl01n8CqoBz82Ijcpwn2IjMoYL6YmRxJCcrlfcI5HoCLIAUd9ivcJWoeZtoe29BkxbWMUD0/IubzObBzIbdamXzl+ccT2rjES9i1jvMTXynx2oY0xF1EuN7DU5WROMSL8zLm4iDNgjczSHsuoja+Hdoz45fsSlgKSAyqEZEPnQZtq5BjyD6bzKjPv8DybbCcgTcrFQuTp2BbPmDPMxOD7GtEJuuD57HJfDzbJcDV/5xV20y6y5kqDXFHIFU1yVSFXNck1hVzTJNcVcl2T3FDIDU1yUyE3NckthdzSJLcVcluTrClkTZPcUcgdTXJPIfc0ybpC1jXJhkI2ckkH94AK0VPk9mTGov6QsZjp5zKyXlpMDWRZhyczAVsMpLQ+y7zyqGCESvcUWWsCDe4usshyB+NP9iGGHqQUwd7juj+ZqrHF6Hpt5TLnw3jSlslx5kd4PhJjSv1PpF3+I3LnbGb/f1CT5dpJqMSnhLy1kO5p415kje9re7jAmOMeLpCBA+uYM3F4B9HzFfJOPe5N7k59Pi0M9upkaKfrV67wMz5L5Hm+sRzzjRNP5fb5RhUa1aXK+6Xqjw+l5ZXs7DNDL+glTmYV+kTLqK+7qCB2Ya5QLXwpfJ39a7wy3hmLqelUIWOe09hlfPwHUO0qDg==</latexit>✓|ḣ± |

Z2⇣

helicity decomposition{

2hL = h+ + ih⇥<latexit sha1_base64="gQHBkznv97wZCS5kf8tYQF1NJNI=">AAAKBniclZbNbtNAEMenpUATvlq4IHGxiJCQkKIkIMEFqfRbaqSWiqSVShT5Y9NYdWLLdtJEUe48AW/QKzfEhQMXeAfehv+OnZCwarzYsnd2dn6zM2trdq3Ac6O4VPq9tHxj5eat26u5/J279+4/WFt/WI/8XmiLmu17fnhqmZHw3K6oxW7sidMgFGbH8sSJdbElx0/6Ioxcv/shHgai0THPu27Ltc0YKn/tMVXIoDY1qYr2bSq/QCsfN+1fUQy5Q4Ki5lqhVCzxZahCORUKlF5H/vpKkz6SQz7Z1GMfXXizySOTItxnVKYSBdA1aARdyHPJcUFjyoPtwUrAwoT2Au9z9M5SbRd9jotpG7N4eEKQBj1LbRzILdYmrZzfmLG9bo4R+5YxDtFaqc8OtDFWJcrkJpa6nMwpRoRvOBcXcQaskVnacxm10Hrox4hfvoewFJAcUCEkGzoP2kQj5wjRJusqM2/zOptsJyAtysVC5MncFq+YM80kz/cVohN0yevZ4bEubEbQ25A9egd6NNM3WDPWIDcVclOT3FLILU1yWyG3NckdhdzRJHcVcleT3FPIPU1yXyH3NckDhTzQJKsKWdUkDxXyUJM8VshjTbKmkDVNsq6Q9UzSwT2hQowY3F/MWDSYMhYzg0xG1kuLqYks6/BiJmCLiZTUZ5lXFhXMUE1uZa0JNLjrSIPlNuZf7ENMPUgpgr3HdX8xVWWL2f+1mcn0p/EkPZPjzI6wPxNjQv1PpB3+InLnbKTff1KT5b8zogJGxpn/QrKnzXuRNX6g7eESc857uEQGDqxjzsThHUTPV8g79bw3uTsN+LQw2atHUztdv/IP7/FZIsvzX8s53zjxlP8936hCvVIsvyxW3r8qbGymZ59VekJP6Tk8vaYN1NcjVBCbPtN3+km/cp9yX3Jfc98S0+WllHlEc1fuxx/krhUR</latexit>

2hR = h+<latexit sha1_base64="JBIxWY+Q6StMh4G3RWL7roSvgTQ=">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</latexit>2
dt d3 x a3 Mpl
LGR + LGCS
2

k
2
|h
|
±
2
a

◆

⌘

k ˙
LGCS = ±LGRa ⇤<latexit sha1_base64="v1KPaVqqNYk/fzUN+jBuXJQ+/0s=">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</latexit>

origin of chiralityih⇥nature of GWs
h = z± h :✓ action for the canonically normalised GW, e<latexit sha1_base64="qWtCxifDuLypE1IoYy/m8k2pR+M=">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</latexit>

1 3
L= a
2
<latexit sha1_base64="CYSn82frxWCXMOhSF5zkcT5LLLE=">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</latexit>



✓

◆✓
k ˙
2
2
Mpl ±
|ḣ± |a ⇤= z±2

1 ė 2=| h± |
2<latexit sha1_base64="Q6bDWBtDKlKziyOBWm4pl7uI7y8=">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</latexit>

✓ 2
k
a2

z̈±
z±

◆

2

k
2
|h± |
2
a

2
e
| h± |

eﬀective mass◆

¨ + 3 c2 H ˙ = 0s<latexit sha1_base64="I8cgfy4JN5vcO23rv3AmkIpY8ws=">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</latexit>c2s =<latexit sha1_base64="9bqz7LWgsIeC+dX9AGoHg2jTs+A=">AAAKMHiclZbLbtNAFIZPS4Gm3FpYsrGIkJCAKglIsEEqvUut1FCRtFIpkS+T1qodW7bTpIr6OH0BnoDHgBViw4INPAL/HDtp3FHjwZY9Z86c71zs0cxYoefGSaXyfWr6xszNW7dnS3N37t67/2B+4WEzDrqRLRp24AXRvmXGwnM7opG4iSf2w0iYvuWJPetkRY7vnYoodoPOx+QsFIe+edRx265tJlAF8y/JphbF9JlqZNA7PBfUpohM6AdUx9iAXkC7T+e4r2oMeo6nBtnIjbB1a75cWazwZahCNRPKlF31YGGmRZ/IoQCxu+SToA4lkD1kE+M+oCpVKITuEJFMZJmQy+MC8ebAdmElYCGzP8H7CL2DTNtBX/qMmbYRxcMTcRVPMxsHcpu1aSvjG2O218UYsG+Z4xlaK/PpQ5vQMbRF3NBSl5M1JcjwLdfiIs+QNbJKO1dRG62HfoL85fsMlgKSAyqCZEPnQZtqZIwIbfpdZeXH/J1NthOQJtViIfM0tsVfzBlVMsf3BbIT1OPv6fNYBzYD6G3IHr3nGXbZN1hzrkEuK+SyJrmikCua5KpCrmqSawq5pkmuK+S6JrmhkBua5KZCbmqSWwq5pUluK+S2JrmjkDua5K5C7mqSDYVsaJJNhWwWkg7uIRVhxOD+ZMai/oixmOkXMnK9tJgaynIdnsyEbDGU0vVZ1lVEhWNUi1u51oQa3HWkwfIx4k/2IUYepBTD3uN1fzK1zRbj87VVyJyO8kl7JudZnOHpWI4p9T+Z+vxH5M55mP3/4Zos586Ayhg5L5wL6Z6W9yLX+L62hx5i5j30UIED64QrcXgH0fMV8U6d9yZ3pz6fFoZ79WBkp+tXzvAunyWKPF9a5nzjxFO9er5RhWZtsfpqsfbhdXlpOTv7zNJjekLP4OkNLWF9rWMFsekr/aI/9Lf0pfSt9KP0MzWdnsqYR5S7Sr//AaOrHgs=</latexit>

X=

P,X
P,X + 2XP,XX(r )2 /2 = ˙ 2 /2<latexit sha1_base64="2FHhoEkinoMR6sHKRUXgFTZv/I8=">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</latexit><latexit sha1_base64="wavQXif+mfSys9o6JRhCnSssoXs=">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</latexit>

h
˙

z̈±
k
2
2
2
2
' H + O("H , ⌘H ) ±H@t (cs ) + (1 + cs )H2z±aMpl ⇤ipreliminary result

1<latexit sha1_base64="/5oRSTIO4sRpUI0zGfkOZRh2SXQ=">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</latexit>

7% amplification..

hR

hL → hGR

hL

h/hGR

end of inflationttheoretical subtleties✓ UV-behaviour of GWs :→ since the GCS includes more spatial derivatives,
it can dominate on small scales..
2
L = MplR+<latexit sha1_base64="G4HS5hIMlpeN95PUDBYfReUJvCc=">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</latexit>⇤e ! ḣ2
RR<latexit sha1_base64="CXNwgjTqCJE906zwHjyPL439Zi4=">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</latexit>✓ ◆3k 2 k ˙ 2
k
˙ 2
h +
ḣ +
h
2
a
a ⇤
a
⇤
2

✓ By tuning the functional form of P [(∇χ)2], GWs can beamplified (as much as you want).
→ but… instabilities in scalar perturbations can show up
2
cs =<latexit sha1_base64="9bqz7LWgsIeC+dX9AGoHg2jTs+A=">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</latexit>

P,X
P,X + 2XP,XX≥0

conclusion✓ We have considered a model with the gravitational ChernSimons term keeping the shift-symmetry of axion.

✓ With the nice form of axion’s kinetic term, the amplitude ofsourced GWs can be larger than that from vacuum fluct.

✓ Despite this interesting results, there are several theoreticalsubtleties which should be overcome..
-- UV behaviour of GWs-- stability of axion

Thank you very much