towards a robust estimate for gravitational leptogenesis...courtesy h.oide towards a robust estimate...

Courtesy H.Oide

Towards a robust estimate for gravitational leptogenesis

Zooming in on Axions in the Early Universe 24/06/2020 @ Virtual Institute at CERN

based on: K. Kamada, J. Kume, Y. Yamada, & J. Yokoyama, JCAP 01 (2020) 016, 1911.02657 [hep-ph]

Kohei Kamada (RESCEU, U Tokyo)

K. Kamada, J. Kume, & Y. Yamada, work in progress (2007.xxxxx?)

Jun’ya Kume (Ph,D student at Tokyo) Yusuke Yamada (PD at Tokyo)

Courtesy H.Oide

The message of this talk1. Gravitational Leptogenesis is yet another possibility to

produce baryon asymmetry through the chiral anomaly implemented in the SM and GR.

2. Appearance of apparent ghost-like degree of freedom as well as UV divergences are the theoretical issues in the scenario.

3. Here we clarify these issues and determine in which case the present BAU can be explained with the scenario.

Courtesy H.Oide

1. Introduction: Baryogenesis and Gravitational Leptogenesis

2. Issues in Gravitational Leptogenesis driven by Axion Inflation

3. Condition for the successful baryogenesis 4. Summary

Courtesy H.Oide

Introduction

Courtesy H.Oide

12 G. Steigman: Primordial Nucleosynthesis

Figure 6. The CBR temperature fluctuation anisotropy spectra for three choices of the baryondensity parameter ωB = 0.018, 0.023, 0.028, in order of increasing height of the first peak. Alsoshown are the WMAP data points.

Figure 7. The normalized likelihood distributions for the baryon density parameter η10 derivedfrom SBBN and the primordial abundance of deuterium (solid curve; see §4.1) and from the CBRusing WMAP data alone (dashed curve). The bottom horizontal axis is the baryon-to-photonratio parameter η10; the top axis is the baryon density parameter ωB = ΩBh2 .

- We live in a matter-antimatter asymmetric Universe. - BBN and CMB can evaluate it quantitatively.

’03 Steigman

23. Big-Bang nucleosynthesis 3

Figure 23.1: The abundances of 4He, D, 3He, and 7Li as predicted by the standardmodel of Big-Bang nucleosynthesis — the bands show the 95% CL range [5].Boxes indicate the observed light element abundances. The narrow vertical bandindicates the CMB measure of the cosmic baryon density, while the wider bandindicates the BBN concordance range (both at 95% CL).

February 8, 2016 19:55

CMBBBN

Particle Data Group 2015

nB

n= (6.09± 0.06) 1010

(Planck 2015) = (6.180± 0.195) 1010

BBN+D ; ’15 Cyburt+

bh2 = 0.018, 0.023, 0.028

Baryon Asymmetry of the Universe

Courtesy H.Oide

Inflation dilutes the preexisting asymmetry.

After inflation before BBN, asymmetry must be generated. matter : anti matter : photon :

~10000000000+1 ~10000000000 ~10000000000

matter : anti matter : photon :

~1 ~0 ~10000000000

annihilationT 100MeV

WMAP team

Courtesy H.Oide

In order to generate baryon asymmetry... Sakharov’s condition is required.

1. B-violation 2. C & CP-violation 3. Deviation from thermal equilibrium

(’67 Sakharov)

Courtesy H.Oide

In order to generate baryon asymmetry... Sakharov’s condition is required.

1. B-violation 2. C & CP-violation 3. Deviation from thermal equilibrium

(’67 Sakharov)

BSM is required!? - Leptogenesis : RH neutrinos - Affleck-Dine : SUSY with B and CP op.

- EW baryogenesis : 1st order EWPT + CP op.

(’85 Fukugita&Yanagida)

(’85 Affleck&Dine, ’95 Dine,Randall&Thomas)

(’85 Kuzmin, Rubakov&Shaposhnikov)

Courtesy H.Oide

Many of baryogenesis models add a new physics beyond the SM that breaks B and/or C&CP.

But the SM ingredients indeed can satisfy the Sakharov’s condition.

Courtesy H.Oide

Many of baryogenesis models add a new physics beyond the SM that breaks B and/or C&CP.

But the SM ingredients indeed can satisfy the Sakharov’s condition.

Chiral anomaly in the SM

µjµB = µjµ

L = Ng

g2

162tr

WµWµ

g2

322Yµ Y µ

(’76 ‘t Hooft)

Gravitational chiral anomaly in the lepton current

rµjµL =

NRL

24(4)2RR, rµj

µB = 0.

<latexit sha1_base64="p1fdEYb1uDEbWs7u9Z0sN42zuxo=">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</latexit>

QB = QL = Ng

NCS

g2

162HY

QL =NRL

24(4)2K0

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(’84 Alvarez-Gaume & Witten)

Courtesy H.Oide

If a mechanism that generates long-range helical hyper gauge fields or gravitational waves works in the early Universe non-thermally, it satisfies the Sakharov’s condition and can generate BAU.

1. B-violation => Chiral anomaly 2. C & CP-violation => Long-range helicity 3. Deviation from thermal equilibrium => non-thermal BG fields

Such a mechanism surely requires physics beyond the SM or GR, but it would open a new window to explore the physics for baryogenesis.

The dynamics of Axion-like particles is the most promising origin.

Courtesy H.Oide

If a mechanism that generates long-range helical hyper gauge fields or gravitational waves works in the early Universe non-thermally, it satisfies the Sakharov’s condition and can generate BAU.

1. B-violation => Chiral anomaly 2. C & CP-violation => Long-range helicity 3. Deviation from thermal equilibrium => non-thermal BG fields

Magnetic field case (’98 Giovannini&Shaposhnikov, ’15 Anber&Sabancilar, ’16 Fujita&KK, ’16 KK&Long)

…B-L conserved => late time dynamics w/ EW sphaleron is important.See Neil, Robert, & Benedict’s talks.

Such a mechanism surely requires physics beyond the SM or GR, but it would open a new window to explore the physics for baryogenesis.

The dynamics of Axion-like particles is the most promising origin.

Courtesy H.Oide

Gravitational Leptogenesis

Lint =N

642

MplRR

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(’06 Alexander, Peskin & Sheikh-Jabbari)

Simple way to generate helical gravitational waves … axion inflation with anomalous coupling to gravity

During inflation helical gravitational waves are generated and lepton asymmetry is generated simultaneously through gravitational chiral anomaly.

Courtesy H.Oide

Gravitational Leptogenesis

Lint =N

642

MplRR

<latexit sha1_base64="G7E13d0Xj/Rz4/BDDb3xU8Je0Xs=">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</latexit>

L = ()R2 4RµR

µ +RµRµ

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LW =

4fTr(WµW

µ)

<latexit sha1_base64="m61cx+cP1VnUobrjoS9SFlaZ+NY=">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</latexit>

(’06 Alexander, Peskin & Sheikh-Jabbari)

Simple way to generate helical gravitational waves … axion inflation with anomalous coupling to gravity

During inflation helical gravitational waves are generated and lepton asymmetry is generated simultaneously through gravitational chiral anomaly.

cf. Several variants - w/ Gauss-Bonnet term

- Chromo-natural inflation

(’19 Kawai & Kim)

(’14 Maleknejad, ’18 Caldwell & Devulder, ’18 Adshead, Long & Sfakianakis)

See Ema & Tomohiro’s talks.

Courtesy H.Oide

Standard (?) estimate for baryon asymmetry(’06 Alexander, Peskin & Sheikh-Jabbari)

1. Parameterize inflation dynamics with and and take them as background.

Hinf ' Hend

<latexit sha1_base64="A9CyD/I8fdlLdtmYE5dDMvx7aDc=">AAACMnicbVDLSgMxFM3UV62vqks3wSq4KjNSUHdFN11WsA9oS8lk7rShSWZMMkIZ+gN+jQs3+ie6E7fu3Zo+QNt6IHA4515y7vFjzrRx3Tcns7K6tr6R3cxtbe/s7uX3D+o6ShSFGo14pJo+0cCZhJphhkMzVkCEz6HhD27GfuMBlGaRvDPDGDqC9CQLGSXGSt38SaXbFsT0lUiZDEe4rZmAe/yrggxG3XzBLboT4GXizUgBzVDt5r/bQUQTAdJQTrRueW5sOilRhlEOo1w70RATOiA9aFkqiQDdSSfXjPCpVQIcRso+afBE/buREqH1UPh2cpxRL3pj8V/PF3M/p9pG60OwEMeElx3bRZwYkHSaJkw4NhEe94cDpoAaPrSEUMXsQZj2iSLU2JZztilvsZdlUj8veqXi1W2pUL6edZZFR+gYnSEPXaAyqqAqqiGKHtETekGvzrPz7nw4n9PRjDPbOURzcL5+AO2gq9U=</latexit>

<latexit sha1_base64="nZ/A+jLhR/0gXR8MhFydW2F9oVg=">AAACF3icbVDLSgMxFL1TX7W+qi7dBIvgqsyIoO6KblxWsA9oh5LJZDqhSWZIMkIZ+hku3OinuBO3Lv0St6btLGzrgcDh3Hu4JydIOdPGdb+d0tr6xuZWebuys7u3f1A9PGrrJFOEtkjCE9UNsKacSdoyzHDaTRXFIuC0E4zupvPOE1WaJfLRjFPqCzyULGIEGyv18n6YGNRPYzYZVGtu3Z0BrRKvIDUo0BxUf6yZZIJKQzjWuue5qfFzrAwjnE4q/UzTFJMRHtKepRILqv18FnmCzqwSoihR9kmDZupfR46F1mMR2E2BTayXZ1Px31kgFi7n2kaLabgUx0TXfs5kmhkqyTxNlHFkEjQtCYVMUWL42BJMFLMfQiTGChNjq6zYprzlXlZJ+6LuXdZvHi5rjduiszKcwCmcgwdX0IB7aEILCCTwDK/w5rw4786H8zlfLTmF5xgW4Hz9AojJoMs=</latexit>

2. Write down the quadratic action for GWs,

NHinf

22M3pl

<latexit sha1_base64="qvaE2XIxsSk1Zj/WWZPirsFX56A=">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</latexit>

A useful parameterization:

S(2)GW =

1

2

Zdd3k

X

A=L,R

z2A(, k)|(hA

k )0|2 k2|hA

k |2

<latexit sha1_base64="TbJs+qDPIirhNHmmRwNQb31vUvA=">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</latexit>

z2A(, k) a2()M2

pl

2

1 Ak

8

<latexit sha1_base64="kIo+ghHw8x6YC17KuscbSf0W0Do=">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</latexit>

R = +1, L = 1

<latexit sha1_base64="h+YeYW+o0urbymlj5M2JsFOtnv0=">AAACMnicbVDLSgMxFM3UV62vUZduglUQ1DIjBXUhFN24cFHFPqAzlkwmbUMzmTHJCGXoD/g1Ltzon+hO3Lp3a9qOYFsPBA7nnEvuPV7EqFSW9WZkZmbn5heyi7ml5ZXVNXN9oyrDWGBSwSELRd1DkjDKSUVRxUg9EgQFHiM1r3sx8GsPREga8lvVi4gboDanLYqR0lLT3HGYDvvo7gaewX37ADr3MfLhr3ql1UO7aeatgjUEnCZ2SvIgRblpfjt+iOOAcIUZkrJhW5FyEyQUxYz0c04sSYRwF7VJQ1OOAiLdZHhNH+5qxYetUOjHFRyqfycSFEjZCzydDJDqyElvIP7recHYz4nUq3WIP7GOap24CeVRrAjHo21aMYMqhIP+oE8FwYr1NEFYUH0QxB0kEFa65Zxuyp7sZZpUjwp2sXB6XcyXztPOsmALbIM9YINjUAKXoAwqAINH8ARewKvxbLwbH8bnKJox0plNMAbj6wdjl6iZ</latexit>

Courtesy H.Oide


3. Solve mode equation

(µAk )

00 +

k2 z00A

zA

(µA

k ) = 0

<latexit sha1_base64="Ph2bG6ZM+F/OQY69d3yApTn8DVk=">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</latexit>

µAk () zA(, k)h

Ak ()

<latexit sha1_base64="0vVCCaD+w8qXY6XA8TfvGNnXhZ8=">AAACRnicbVDLSgMxFL1T3/VVdekmWAQLUmakYN352LisYFVw6pBJb21oMjMmmUId+it+jQs3uvcn3Em3po+Fth4IHM45l9x7wkRwbVz308nNzS8sLi2v5FfX1jc2C1vbNzpOFcM6i0Ws7kKqUfAI64YbgXeJQipDgbdh52Lo33ZRaR5H16aXYEPSx4i3OKPGSkGh6ss0yPxQkk7/4ezAR0NLxMenlHfJczAWDjsl0p4OBYWiW3ZHILPEm5AiTFALCgO/GbNUYmSYoFrfe25iGhlVhjOB/byfakwo69BHvLc0ohJ1Ixtd2Cf7VmmSVqzsiwwZqb8nMiq17snQJiU1bT3tDcV/vVD++TnTdrU2NqfWMa1qI+NRkhqM2HibViqIicmwU9LkCpkRPUsoU9weRFibKsqMbT5vm/Kme5klN0dlr1I+uaoUT88nnS3DLuzBAXhwDKdwCTWoA4MXeIV3+HDenC/n2xmMozlnMrMDf5CDH3JbsLE=</latexit>

with assuming and asymptotic plane wave at early epochs|k| 1

<latexit sha1_base64="Ll7106lZZ2bQgLUX0KlqHt+Bvfk=">AAACInicbVDLSgMxFM3UV62v8bFzEyyCqzIjBXVXdOOyQl/QGUomc6cNzTxIMkKd9l9cuNFPcSeuBP/DrWk7C9t6IOFwzr3k5HgJZ1JZ1pdRWFvf2Nwqbpd2dvf2D8zDo5aMU0GhSWMei45HJHAWQVMxxaGTCCChx6HtDe+mfvsRhGRx1FCjBNyQ9CMWMEqUlnrmyXjoNAagCHb0NcYO59jumWWrYs2AV4mdkzLKUe+ZP44f0zSESFFOpOzaVqLcjAjFKIdJyUklJIQOSR+6mkYkBOlms/QTfK4VHwex0CdSeKb+3chIKOUo9PRkSNRALntT8V/PCxdezqSONgB/KY4Krt2MRUmqIKLzNEHKsYrxtC/sMwFU8ZEmhAqmP4TpgAhClW61pJuyl3tZJa3Lil2t3DxUy7XbvLMiOkVn6ALZ6ArV0D2qoyai6Ak9o1f0ZrwY78aH8TkfLRj5zjFagPH9CwpipA4=</latexit>

µAk () =

1pkexpiki exp

A

32

W,3/2

i

r4 32

64k

!

<latexit sha1_base64="TzFtp4fQK2mEnGUL60JN1mx1xok=">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</latexit>

iA

p256/2 3

<latexit sha1_base64="wnpOWNJn41Og2dbjnUx0fNy3a5A=">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</latexit>

4. Calculate the 0th-component of the Chern-Simons current

K0(f) =1

2a4f

Z

k<af

d3k

(2)3kX

A=R,L

A

z2A

"|µA

k0|2

k2

z0AzA

2!|µA

k |2 z0AzA

(µAkµ

Ak

0+ h.c.)

#

<latexit sha1_base64="9ShPu3lMmY77LHHMN9avxiL4IGQ=">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</latexit>

:

<latexit sha1_base64="NbH9OHhD1hsdfs7FehAcxLYVNTI=">AAACFXicbVC7SgNBFJ2NrxhfUUubwSBYhV0J+KiCNhYWEcxDkiXMzt4kQ2Zml5lZISz5Cgsb/RQ7sbX2S2ydJFuYxAMDh3PP5Z45QcyZNq777eRWVtfWN/Kbha3tnd294v5BQ0eJolCnEY9UKyAaOJNQN8xwaMUKiAg4NIPhzWTefAKlWSQfzCgGX5C+ZD1GibHSY+fOWkNyhbvFklt2p8DLxMtICWWodYs/nTCiiQBpKCdatz03Nn5KlGGUw7jQSTTEhA5JH9qWSiJA++k08BifWCXEvUjZJw2eqn83UiK0HonAOgUxA704m4j/zgIxdznVNtoAwoU4pnfhp0zGiQFJZ2l6CccmwpOKcMgUUMNHlhCqmP0QpgOiCDW2yIJtylvsZZk0zspepXx5XylVr7PO8ugIHaNT5KFzVEW3qIbqiCKBntErenNenHfnw/mcWXNOtnOI5uB8/QLr5p9W</latexit>

cutoff scale for GW production'

162ainf

Hf

Mpl

2

Hinf

4

H3inf .

<latexit sha1_base64="SYQ2JBR897iwoVr8rbqOwTYDFqM=">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</latexit>

Courtesy H.Oide


5. Through the gravitational chiral anomaly, lepton asymmetry is produced.

µ :

<latexit sha1_base64="97dZA+gBvIepvmWUX6saHAp2p88=">AAACEXicbVDLSgMxFL1TX7W+qi7dBIvgqsxIwceq6MZlBacttEPJZNI2NMkMSUYoQ7/BhRv9FHfi1i/wS9yatrOwrQcCh3Pu5Z6cMOFMG9f9dgpr6xubW8Xt0s7u3v5B+fCoqeNUEeqTmMeqHWJNOZPUN8xw2k4UxSLktBWO7qZ+64kqzWL5aMYJDQQeSNZnBBsr+V2R3qBeueJW3RnQKvFyUoEcjV75pxvFJBVUGsKx1h3PTUyQYWUY4XRS6qaaJpiM8IB2LJVYUB1ks7ATdGaVCPVjZZ80aKb+3ciw0HosQjspsBnqZW8q/uuFYuFypm20IY2W4pj+VZAxmaSGSjJP0085MjGa1oMipigxfGwJJorZDyEyxAoTY0ss2aa85V5WSfOi6tWq1w+1Sv0276wIJ3AK5+DBJdThHhrgAwEGz/AKb86L8+58OJ/z0YKT7xzDApyvXweRnc8=</latexit>

cutoff scale for chiral anomaly

6. After reheating EW sphaleron transfers lepton asymmetry to baryon asymmetry

nL(f) =NRL

24(4)2af K

0(f)k<µaf

'

20484

Hinf

Mpl

2 µ

Hinf

4

H3inf

<latexit sha1_base64="dNRaSAr7siHG1yB3TDYBuZta4kw=">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</latexit>

nB = 28

79nL

af

a

3=

7

404484

Hinf

Mpl

2 µ

Hinf

4

H3inf

af

a

3

<latexit sha1_base64="pv47c9hyqjY6bqDe0vU1mS3nMFA=">AAAC/nicjVJNa9swGJa9ry77SrvjLmJh0B0W7NSQ9FAo3aWHDTpo0kKdGFmRY1FJNpJcCKpgl+2n9FZ23V/Zv9ht18lJFuZ0h70gePS8H3r0SGnJqNJB8MPz791/8PDR1uPWk6fPnr9ob++MVFFJTIa4YIU8T5EijAoy1FQzcl5KgnjKyFl6+b7On10RqWghTvW8JGOOZoJmFCPtqKT9UyRH8AC+izOJsOkNrOnvWyiSDzBmJNO7Sx4lMUc6l9xk1hpkY0lnuX472TuAy4K+NVEQRQMYl3QSWRif5kSjxozj9Qwq6ikf1/uS2fXEXqMn5pXdbPxTGcFmYrL3X4qTdifoBouAd0G4Ah2wipOk/SueFrjiRGjMkFIXYVDqsUFSU8yIbcWVIiXCl2hGLhwUiBM1NouXsfCNY6YwK6RbQsMF+3eHQVypOU9dZS1XbeZq8p+5lDdONspJy8l0Q47OBmNnTllpIvBSTVYxqAtY/wU4pZJgzeYOICypuxDEOXL2afdjWs6pcNOXu2DU64ZRd/9T1Dk8Wnm2BV6B12AXhKAPDsExOAFDgL2Rd+198b76n/0b/9b/tiz1vVXPS9AI//tv5ir3Fw==</latexit>

Courtesy H.Oide


5. Through the gravitational chiral anomaly, lepton asymmetry is produced.

µ :

<latexit sha1_base64="97dZA+gBvIepvmWUX6saHAp2p88=">AAACEXicbVDLSgMxFL1TX7W+qi7dBIvgqsxIwceq6MZlBacttEPJZNI2NMkMSUYoQ7/BhRv9FHfi1i/wS9yatrOwrQcCh3Pu5Z6cMOFMG9f9dgpr6xubW8Xt0s7u3v5B+fCoqeNUEeqTmMeqHWJNOZPUN8xw2k4UxSLktBWO7qZ+64kqzWL5aMYJDQQeSNZnBBsr+V2R3qBeueJW3RnQKvFyUoEcjV75pxvFJBVUGsKx1h3PTUyQYWUY4XRS6qaaJpiM8IB2LJVYUB1ks7ATdGaVCPVjZZ80aKb+3ciw0HosQjspsBnqZW8q/uuFYuFypm20IY2W4pj+VZAxmaSGSjJP0085MjGa1oMipigxfGwJJorZDyEyxAoTY0ss2aa85V5WSfOi6tWq1w+1Sv0276wIJ3AK5+DBJdThHhrgAwEGz/AKb86L8+58OJ/z0YKT7xzDApyvXweRnc8=</latexit>

cutoff scale for chiral anomaly

6. After reheating EW sphaleron transfers lepton asymmetry to baryon asymmetry

nL(f) =NRL

24(4)2af K

0(f)k<µaf

'

20484

Hinf

Mpl

2 µ

Hinf

4

H3inf

<latexit sha1_base64="dNRaSAr7siHG1yB3TDYBuZta4kw=">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</latexit>

nB = 28

79nL

af

a

3=

7

404484

Hinf

Mpl

2 µ

Hinf

4

H3inf

af

a

3

<latexit sha1_base64="pv47c9hyqjY6bqDe0vU1mS3nMFA=">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</latexit>

Present BAU can be explained for the instant reheating with ..?µ ' Mpl

<latexit sha1_base64="eFHm4/kEsje1BO+jw9wsR6cajUQ=">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</latexit>

Courtesy H.Oide

Issues in the estimate of the baryon asymmetry

Courtesy H.Oide

Do you think the estimate I explained is reasonable?

😕

Courtesy H.Oide

1. At early times , inevitably one of the helicity modes satisfies . => Appearance of ghost mode.

! 1

<latexit sha1_base64="bsiRk3ReebXYlELn+04GLEjgHzg=">AAACJnicbVBNS8NAFNz4WetX1IvgZbEIXiyJFNSb6MVjBatCU8pm+9Iu3WzC7osSgv4aD170p3gT8ea/8Oq25qDVgYVh5j123oSpFAY9792Zmp6ZnZuvLFQXl5ZXVt219UuTZJpDiycy0dchMyCFghYKlHCdamBxKOEqHJ6O/Ksb0EYk6gLzFDox6ysRCc7QSl13MwBkNNCiP0CmdXJL9wKhIsy7bs2re2PQv8QvSY2UaHbdz6CX8CwGhVwyY9q+l2KnYBoFl3BXDTIDKeND1oe2pYrFYDrF+II7umOVHo0SbZ9COlZ/bhQsNiaPQzsZMxyYSW8k/uuF8a+fC2OjDaA3EQejw04hVJohKP6dJsokxYSOOqM9oYGjzC1hXAt7EOUDphlH22zVNuVP9vKXXO7X/Ub96LxROz4pO6uQLbJNdolPDsgxOSNN0iKc3JMH8kSenUfnxXl13r5Hp5xyZ4P8gvPxBUOaplQ=</latexit>

z2A < 0

<latexit sha1_base64="0t0XYp4k8xyTPFAtkXyi4Wf87Oc=">AAACFXicbVC7TsMwFL3hWcqrwMhiUSExVUlVCZAYCiyMRaIP1IbKcZzWqp1EtoNUon4FAwt8ChtiZeZLWHHbDLTlSJaOzrlX9/h4MWdK2/a3tbS8srq2ntvIb25t7+wW9vYbKkokoXUS8Ui2PKwoZyGta6Y5bcWSYuFx2vQG12O/+UilYlF4p4cxdQXuhSxgBGsj3T91Lx/K6ALZ3ULRLtkToEXiZKQIGWrdwk/Hj0giaKgJx0q1HTvWboqlZoTTUb6TKBpjMsA92jY0xIIqN50EHqFjo/goiKR5oUYT9e9GioVSQ+GZSYF1X817Y/FfzxMzl1NlovWpPxdHB2duysI40TQk0zRBwpGO0Lgi5DNJieZDQzCRzHwIkT6WmGhTZN405cz3skga5ZJTKZ3fVorVq6yzHBzCEZyAA6dQhRuoQR0ICHiGV3izXqx368P6nI4uWdnOAczA+voF3wietQ==</latexit>

2. Baryon asymmetry/Gravitational waves UV divergent. Is it physical?

👻Remember Leila’s talk

Courtesy H.Oide

Is ghost dangerous? zA = 0

<latexit sha1_base64="D6jZeYdQ/HvPSkyLG5osVsh7xlw=">AAACEXicbVDLSsNAFJ3UV62vqks3g0VwVRIpqAuh6sZlBdMW2lAmk5t26GQSZiZCDf0GF270U9yJW7/AL3HrtM3Cth4YOJxzL/fM8RPOlLbtb6uwsrq2vlHcLG1t7+zulfcPmipOJQWXxjyWbZ8o4EyAq5nm0E4kkMjn0PKHtxO/9QhSsVg86FECXkT6goWMEm0k96l3fWX3yhW7ak+Bl4mTkwrK0eiVf7pBTNMIhKacKNVx7ER7GZGaUQ7jUjdVkBA6JH3oGCpIBMrLpmHH+MQoAQ5jaZ7QeKr+3chIpNQo8s1kRPRALXoT8V/Pj+YuZ8pEG0CwEEeHF17GRJJqEHSWJkw51jGe1IMDJoFqPjKEUMnMhzAdEEmoNiWWTFPOYi/LpHlWdWrVy/tapX6Td1ZER+gYnSIHnaM6ukMN5CKKGHpGr+jNerHerQ/rczZasPKdQzQH6+sX6zKdvg==</latexit>

= 8

k, kphys =

k

a=

8Hinf

<latexit sha1_base64="uCkkJnmOflKE9QYZ3ovXTAYBPHY=">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</latexit>

at

If the physical momentum at that time is larger than the cutoff scale of the theory, the ghost might not be physical.

(’05 Alexander & Martin)

It just gives the upper bound of : || < 8Hinf

Mpl

<latexit sha1_base64="CO//odVbko1gnEuAmRVvnsZ/AU8=">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</latexit>

<latexit sha1_base64="hylDeY8CUP0GQmPoeBNlm/3dqig=">AAACEnicbVDLSgMxFM3UV62vqks3wSK4KjNSUHdFNy4r9CG0Q8lk7rSxSWZIMkIZ+g8u3OinuBO3/oBf4ta0nYVtPRA4nHMv9+QECWfauO63U1hb39jcKm6Xdnb39g/Kh0dtHaeKQovGPFYPAdHAmYSWYYbDQ6KAiIBDJxjdTv3OEyjNYtk04wR8QQaSRYwSY6V2rzkEQ/rlilt1Z8CrxMtJBeVo9Ms/vTCmqQBpKCdadz03MX5GlGGUw6TUSzUkhI7IALqWSiJA+9ks7QSfWSXEUazskwbP1L8bGRFaj0VgJwUxQ73sTcV/vUAsXM60jTaEcCmOia78jMkkNSDpPE2UcmxiPO0Hh0wBNXxsCaGK2Q9hOiSKUGNbLNmmvOVeVkn7ourVqtf3tUr9Ju+siE7QKTpHHrpEdXSHGqiFKHpEz+gVvTkvzrvz4XzORwtOvnOMFuB8/QJjMp6T</latexit>

N < 1.1 1071/2

Hinf

105Mpl

1

<latexit sha1_base64="XmMix0EDrhEQbOtjqzmcUN8zjc0=">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</latexit>

For the slow-roll inflation:

👻Remember Leila’s talk

Courtesy H.Oide

UV divergence?1. An approach is to say ..

The divergence is physical, which comes from unknown UV physics. The cut-off scale of the chiral gravitational effect is taken to be the right-handed neutrino mass, , with type-I sea-saw being in mind.

µ = MR

<latexit sha1_base64="qGsGW4A7Gy4hjVuGfxoevKn/KCY=">AAACIHicbVDLSsNAFJ3UV62vWJduBovgqiRSUBdC0Y0boYp9QBPCZDJph84kYWYiltBfceFGP8WduNQfceukzcK2Hhg4nHMv98zxE0alsqwvo7Syura+Ud6sbG3v7O6Z+9WOjFOBSRvHLBY9H0nCaETaiipGeokgiPuMdP3Rde53H4mQNI4e1DghLkeDiIYUI6Ulz6w6PIWX8NZzOFJDwbP7iWfWrLo1BVwmdkFqoEDLM3+cIMYpJ5HCDEnZt61EuRkSimJGJhUnlSRBeIQGpK9phDiRbjbNPoHHWglgGAv9IgWn6t+NDHEpx9zXk3lCuejl4r+ez+cuZ1JHG5JgIY4Kz92MRkmqSIRnacKUQRXDvC0YUEGwYmNNEBZUfwjiIRIIK91pRTdlL/ayTDqndbtRv7hr1JpXRWdlcAiOwAmwwRloghvQAm2AwRN4Bq/gzXgx3o0P43M2WjKKnQMwB+P7F+CHo4A=</latexit>

2. The other approach is to say ..

We should renormalize it!!!

MR < 1016GeV

<latexit sha1_base64="uQvJLlre1fqK5h9fxPg7BkLOkRE=">AAACL3icbVDLSgMxFM3UV62vUZduokVwVWak+AAXRRe6EarYB7S1ZNLbNjSZGZKMUIZZ+zUu3OiniBtx6xe4NX0ItvVA4HDOveTc44WcKe0471Zqbn5hcSm9nFlZXVvfsDe3yiqIJIUSDXggqx5RwJkPJc00h2oogQiPQ8XrXQz8ygNIxQL/TvdDaAjS8VmbUaKN1LR3r5t1QXRXivg2OXOd+9g9SvCvdAnlpGlnnZwzBJ4l7phk0RjFpv1dbwU0EuBryolSNdcJdSMmUjPKIcnUIwUhoT3SgZqhPhGgGvHwlATvG6WF24E0z9d4qP7diIlQqi88MznIqKa9gfiv54mJn2NlonWhNRVHt08aMfPDSINPR2naEcc6wIPycItJoJr3DSFUMnMQpl0iCdWm4oxpyp3uZZaUD3NuPnd6k88WzsedpdEO2kMHyEXHqICuUBGVEEWP6Am9oFfr2XqzPqzP0WjKGu9sowlYXz/ymqm0</latexit>

=> constraint:

Courtesy H.Oide

How to renormalize the UV divergence

hRR()i = 1

2a4@

Z

k<af

d3k

(2)3kX

A=R,L

A

z2A

"|µA

k0|2

k2

z0AzA

2!|µA

k |2 z0AzA

(µAkµ

Ak

0+ h.c.)

#

<latexit sha1_base64="Gnb0/T51mTCk/f6ANblyAU7CP5o=">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</latexit>

with µAk () =

1pkexpiki exp

A

32

W,3/2

i

r4 32

64k

!


hRR()i =3H6

inf

2M2pl

"

Hinf

4

+ 3

Hinf

2

10 log

2

Hinf

+

39

2 10

#+O

2

<latexit sha1_base64="phSow7PX49mqewypi/icUqibf80=">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</latexit>

# by using the Mellin-Barens representation

W,µ(z) =ez/2

2i

Z i1

i1dsdzs

(s )(s µ+ 1/2)(s+ µ+ 1/2)

( µ+ 1/2)(µ + 1/2)

<latexit sha1_base64="EYOcx8jXt0476b7ucNXvqR12QUE=">AAACzHicbZFNbxMxEIa9y1cJXykcuVgEpFQl6W5UqfRQqYIDiAMqomkqdZOV1+ttrNhey/YiJZav/Cz+B7+EK07iVm3KSJbefWZe73imkIxqkyR/ovje/QcPH209bj15+uz5i/b2yzNdNwqTIa5Zrc4LpAmjggwNNYycS0UQLxgZFbNPy/zoJ1Ga1uLUzCUZc3QpaEUxMh7l7d+j3GYzJCV6n/HGdRc78AhmlULYkontLfYGztkBzCSF1MGMCpPbHl2KyszdxF5LWGpYLiY6mLPPiHPU1b315TswgJ4nvNlN9wY30O4Vcle+YLuuDdh/hsy6HObtTtJPVgHvijSIDghxkrf/ZmWNG06EwQxpfZEm0owtUoZiRlwrazSRCM/QJbnwUiBO9Niu5uzgO09KWNXKH2Hgit50WMS1nvPCV3Jkpnozt4T/zRX81p+t9q1NSbnRjqk+jC0VsjFE4HU3VcOgqeFys7CkimDD5l4grKh/EMRT5Jdh/P5bflLp5lzuirNBP93vH37f7xx/DDPbAq/BG9AFKTgAx+ALOAFDgKO30dfoR3Qaf4tNbGO3Lo2j4HkFbkX86x8I6trS</latexit>

1p256/2 3

'

16

<latexit sha1_base64="cjNHlJLtY6G4vEDUI2uu05Ft1Uc=">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</latexit>

(cf. ’18 Adshead+, ’19 Ballardini+ )

Courtesy H.Oide

How to renormalize the UV divergence

hRR()i = 1

2a4@

Z

k<af

d3k

(2)3kX

A=R,L

A

z2A

"|µA

k0|2

k2

z0AzA

2!|µA

k |2 z0AzA

(µAkµ

Ak

0+ h.c.)

#

<latexit sha1_base64="Gnb0/T51mTCk/f6ANblyAU7CP5o=">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</latexit>

with µAk () =

1pkexpiki exp

A

32

W,3/2

i

r4 32

64k

!


hRR()i =3H6

inf

2M2pl

"

Hinf

4

+ 3

Hinf

2

10 log

2

Hinf

+

39

2 10

#+O

2

<latexit sha1_base64="phSow7PX49mqewypi/icUqibf80=">AAADRniclVJNb9NAEF27fJTw0RSOXFZESKkiItuNKD0gVXDpAURBTVspm0Tj9dpZZde2djdIkbV/hV/DgQvc+RPcUK9sEoOaFAkxkldP782MZ95uXAquTRB89/ytGzdv3d6+07h77/6DnebuwzNdzBRlfVqIQl3EoJngOesbbgS7KBUDGQt2Hk9fL/Tzj0xpXuSnZl6yoYQs5ymnYBw13vUOiIA8Ewx/wBUxXCQOWdwmzMAeJmqlvcQkVUCr/eMxkWAmSlY8T+3oua1IyUcRfvuHL4UdRfZ3KzKFsgSLiWCpGSzP9qoVeeOGTMBW6y0tUTybmL1Rr7OP/yc/ehYGRBTZ1ZroH0W4U691aKvILhpkICWs1GGnIhQEfmfrlqcT58koqmvHzVbQDZaBr4OwBi1Ux8m4eUmSgs4kyw0VoPUgDEozrEAZTgWzDTLTrAQ6hYwNHMxBMj2sljds8VPHJDgtlPtyg5fs1YoKpNZzGbvMxZ56U1uQf9ViufbnSrvRJizZGMekL4bOuXJmWE5X06QzgU2BF28KJ1wxasTcAaCKu4UwnYBz1riX13BOhZu+XAdnUTfsdQ/f91pHr2rPttFj9AS1UYgO0BE6Rieoj6j3yfvsffW++V/8H/5P/3KV6nt1zSO0FlvoF69dEZo=</latexit>

# by using the Mellin-Barens representation

W,µ(z) =ez/2

2i

Z i1

i1dsdzs

(s )(s µ+ 1/2)(s+ µ+ 1/2)

( µ+ 1/2)(µ + 1/2)

<latexit sha1_base64="EYOcx8jXt0476b7ucNXvqR12QUE=">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</latexit>

(cf. ’18 Adshead+, ’19 Ballardini+ )

“Minimal subtraction” 1p256/2 3

'

16

<latexit sha1_base64="cjNHlJLtY6G4vEDUI2uu05Ft1Uc=">AAACXnicbVHBbtNAEN2YUkqgNMAFqZcVERIXgh1KW24VXHpMpaatVIdovB43q+zazu64UrTyb/EvlXrgAr/AlU3iQ5t2pJWe3nujmXmblEpaCsObVvBk4+nms63n7Rcvt1/tdF6/ObNFZQQORaEKc5GARSVzHJIkhRelQdCJwvNk+mOhn1+jsbLIT2le4kjDVS4zKYA8Ne4MXExSpcjjKZQl1DzGWSWveZwZEC6qXWxnhlz/6/7n+HSCBD/7/NOX2vus1DhrfCupdtF+Pe50w164LP4QRA3osqYG486/OC1EpTEnocDayygsaeTAkBQK63ZcWSxBTOEKLz3MQaMdueXlNf/gmZRnhfEvJ75k73Y40NbOdeKdGmhi17UF+aiW6HuTnfWrTTBdW4eyw5GTeVkR5mK1TVYpTgVfZM1TaVCQmnsAwkh/EBcT8IGR/5G2Typaz+UhOOv3or3et5O97tH3JrMttsves48sYgfsiB2zARsywX6xW/aH/W39DjaD7WBnZQ1aTc9bdq+Cd/8B7ta40A==</latexit>

Courtesy H.Oide

UV divergence can be removed by the following center terms,

EOM for inflaton :

N642Mpl

hRRiren

<latexit sha1_base64="0HI++S/x7g1huUij3wYBkLW7OAo=">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</latexit>

(cf. ’07 Fischler & Paban, ’19 Ballardini+ )

+ · · ·

<latexit sha1_base64="s1zRjbD13127Xa7M4yFy6pGJ2WM=">AAACE3icbVDLSgMxFL1TX7W+qi7dBIsgCGVGCuqu6MZlBfuAdiiZTKYNTSZDkhHK0I9w4UY/xZ249QP8Erem7Sxs64HA4Zx7uScnSDjTxnW/ncLa+sbmVnG7tLO7t39QPjxqaZkqQptEcqk6AdaUs5g2DTOcdhJFsQg4bQeju6nffqJKMxk/mnFCfYEHMYsYwcZK7YseCaXR/XLFrbozoFXi5aQCORr98k8vlCQVNDaEY627npsYP8PKMMLppNRLNU0wGeEB7VoaY0G1n83iTtCZVUIUSWVfbNBM/buRYaH1WAR2UmAz1MveVPzXC8TC5UzbaEMaLsUx0bWfsThJDY3JPE2UcmQkmhaEQqYoMXxsCSaK2Q8hMsQKE2NrLNmmvOVeVknrsurVqjcPtUr9Nu+sCCdwCufgwRXU4R4a0AQCI3iGV3hzXpx358P5nI8WnHznGBbgfP0CEbye7w==</latexit>

=@2+ V 0 =

N642Mpl

hRRi Lct

<latexit sha1_base64="n6nDrfmImmqLRXYSEujICZbdl/U=">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</latexit>

Lct =(a1R3gµ + a2R(RR)g

µ + a3(RR)R

µ)@µ@

+ (a4R2 + a5R

µRµ)(@µ@) + (a6Rgµ + a7Rµ)(@µ@)

<latexit sha1_base64="MC/QX0YKeCbmvUdudorM423UmT4=">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</latexit>

Courtesy H.Oide

If we take the minimal subtraction, the resultant BAU is

nB =7

404484

39

2 10

Hinf

Mpl

2

H3inf

af

a

3

<latexit sha1_base64="iHC8ZtfB6h0RzOepndIbsI4VFY8=">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</latexit>

For the instant reheating, we obtainnB

s' 8.3 1028

8Hinf/Mpl

Hinf

105Mpl

9/2

<latexit sha1_base64="fL7yq2j1EHOJifcmxVTWgTOqRMY=">AAACq3icbVHLThsxFPUMfdD0QVqW3ViNKsGCMBOoGnaIbthUoioBVBxGHudOxsLzqH2nUmT5U/phfEm3dSYBNaFXsnR87j328XFaK2kwiu6CcOPJ02fPN190Xr56/War+/bdhakaLWAkKlXpq5QbULKEEUpUcFVr4EWq4DK9/TLvX/4CbWRVnuOshnHBp6XMpODoqaT7m2WaC1smJ84aR5mRBfykw/4BZeihoXF0Y/cGQ99qB9l5DsidHdLThBUcc11YWWZu/+vDtlbO3auZggx3FtJVgbPtyZ8cXVUyLac57t7Yo/2BS7q9qB+1RR+DeAl6ZFlnSfcPm1SiKaBEobgx13FU49hyjVIocB3WGKi5uOVTuPaw5N7k2LYxOvrRMxOaVdqvEmnL/quwvDBmVqR+cm7YrPfm5H97abFyszXeWg6TNTuYDcc+mrpBKMXCTdYoihWdfxydSA0C1cwDLrT0D6Ii5z5Y9N/b8UnF67k8BheDfnzYP/p22Ds+WWa2Sd6TD2SHxOQzOSan5IyMiAg2gt1gEByEe+H38EfIFqNhsNRsk5UK4S/wRdP5</latexit>

It is hard to explain the present BAU ...

😢

Courtesy H.Oide

If we take the minimal subtraction, the resultant BAU is

nB =7

404484

39

2 10

Hinf

Mpl

2

H3inf

af

a

3

<latexit sha1_base64="iHC8ZtfB6h0RzOepndIbsI4VFY8=">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</latexit>

For the instant reheating, we obtainnB

s' 8.3 1028

8Hinf/Mpl

Hinf

105Mpl

9/2

<latexit sha1_base64="fL7yq2j1EHOJifcmxVTWgTOqRMY=">AAACq3icbVHLThsxFPUMfdD0QVqW3ViNKsGCMBOoGnaIbthUoioBVBxGHudOxsLzqH2nUmT5U/phfEm3dSYBNaFXsnR87j328XFaK2kwiu6CcOPJ02fPN190Xr56/War+/bdhakaLWAkKlXpq5QbULKEEUpUcFVr4EWq4DK9/TLvX/4CbWRVnuOshnHBp6XMpODoqaT7m2WaC1smJ84aR5mRBfykw/4BZeihoXF0Y/cGQ99qB9l5DsidHdLThBUcc11YWWZu/+vDtlbO3auZggx3FtJVgbPtyZ8cXVUyLac57t7Yo/2BS7q9qB+1RR+DeAl6ZFlnSfcPm1SiKaBEobgx13FU49hyjVIocB3WGKi5uOVTuPaw5N7k2LYxOvrRMxOaVdqvEmnL/quwvDBmVqR+cm7YrPfm5H97abFyszXeWg6TNTuYDcc+mrpBKMXCTdYoihWdfxydSA0C1cwDLrT0D6Ii5z5Y9N/b8UnF67k8BheDfnzYP/p22Ds+WWa2Sd6TD2SHxOQzOSan5IyMiAg2gt1gEByEe+H38EfIFqNhsNRsk5UK4S/wRdP5</latexit>

It is hard to explain the present BAU ...But is there any reason we shall employ the minimal subtraction? The renormalization condition that fixes the finite term that remains after renormalization cannot be determined.

Without the knowledge of the UV completion, we cannot exclude the possibility that we can keep the large finite term.

😢

Courtesy H.Oide

Conditions for the successful baryogenesis

Courtesy H.Oide

Adopt the lepton asymmetry just after inflation:

nL(f) '

20484

Hinf

Mpl

2 µ

Hinf

4

H3inf

<latexit sha1_base64="Opp53K12iCH0k8pYxeB6zcodR6I=">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</latexit>

The cosmic history:

End of inflation Reheating Mat-Rad

equality Present

Hf ' Hinf

<latexit sha1_base64="ATV4v6eZvVh8EXmlAS1av3re0Dc=">AAACMHicbVDLSgMxFM3UV62vUZduQovgqsxIQd0V3XRZwT6gLSWT3mlDk5kxyQhl6N6vceFGP0VX4tYfcGumHdC2Hggczr2XnHO8iDOlHefdyq2tb2xu5bcLO7t7+wf24VFThbGk0KAhD2XbIwo4C6ChmebQjiQQ4XFoeeObdN56AKlYGNzpSQQ9QYYB8xkl2kh9u1jrdwXRIykSf4q7igm4x78aC/xp3y45ZWcGvErcjJRQhnrf/u4OQhoLCDTlRKmO60S6lxCpGeUwLXRjBRGhYzKEjqEBEaB6ySzLFJ8aZYD9UJoXaDxT/14kRCg1EZ7ZTD2q5Vkq/jvzxMLPiTLWRjBYsqP9y55JHcUaAjp348cc6xCn7eEBk0A1nxhCqGQmEKYjIgnVpuOCacpd7mWVNM/LbqV8dVspVa+zzvLoBBXRGXLRBaqiGqqjBqLoET2hF/RqPVtv1of1OV/NWdnNMVqA9fUDKzGq8A==</latexit>

T = TR

<latexit sha1_base64="6anX4vLinp2i2UXkif+dCyX0CJo=">AAACEXicbVDLSsNAFJ3UV62vqks3g0VwVRIpqAuh6MZllaYttKFMJjft0MkkzEyEEvoNLtzop7gTt36BX+LWaZuFbT0wcDjnXu6Z4yecKW3b31ZhbX1jc6u4XdrZ3ds/KB8etVScSgoujXksOz5RwJkAVzPNoZNIIJHPoe2P7qZ++wmkYrFo6nECXkQGgoWMEm0kt3nT7D/2yxW7as+AV4mTkwrK0eiXf3pBTNMIhKacKNV17ER7GZGaUQ6TUi9VkBA6IgPoGipIBMrLZmEn+MwoAQ5jaZ7QeKb+3chIpNQ48s1kRPRQLXtT8V/PjxYuZ8pEG0KwFEeHV17GRJJqEHSeJkw51jGe1oMDJoFqPjaEUMnMhzAdEkmoNiWWTFPOci+rpHVRdWrV64dapX6bd1ZEJ+gUnSMHXaI6ukcN5CKKGHpGr+jNerHerQ/rcz5asPKdY7QA6+sXA3KdzQ==</latexit>

EOSw

<latexit sha1_base64="oTxWFQduV47F1iZIeUu187mprZc=">AAACDXicbVDLSgMxFM3UV62vqks3wSK4KjMiqLuiG5ct2Ae0Q8lk7rShSWZIMkoZ+gUu3OinuBO3foNf4ta0nYVtPRA4nHMv9+QECWfauO63U1hb39jcKm6Xdnb39g/Kh0ctHaeKQpPGPFadgGjgTELTMMOhkyggIuDQDkZ3U7/9CEqzWD6YcQK+IAPJIkaJsVLjqV+uuFV3BrxKvJxUUI56v/zTC2OaCpCGcqJ113MT42dEGUY5TEq9VENC6IgMoGupJAK0n82CTvCZVUIcxco+afBM/buREaH1WAR2UhAz1MveVPzXC8TC5UzbaEMIl+KY6NrPmExSA5LO00QpxybG02pwyBRQw8eWEKqY/RCmQ6IINbbAkm3KW+5llbQuqt5l9aZxWand5p0V0Qk6RefIQ1eohu5RHTURRYCe0St6c16cd+fD+ZyPFpx85xgtwPn6BY+QnIY=</latexit>

No late-time entropy production

Baryon-to-entropy ratio is fixed here

Courtesy H.Oide

Adopt the lepton asymmetry just after inflation:

nL(f) '

20484

Hinf

Mpl

2 µ

Hinf

4

H3inf

<latexit sha1_base64="Opp53K12iCH0k8pYxeB6zcodR6I=">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</latexit>

The cosmic history:

End of inflation Reheating Mat-Rad

equality Present

Hf ' Hinf

<latexit sha1_base64="ATV4v6eZvVh8EXmlAS1av3re0Dc=">AAACMHicbVDLSgMxFM3UV62vUZduQovgqsxIQd0V3XRZwT6gLSWT3mlDk5kxyQhl6N6vceFGP0VX4tYfcGumHdC2Hggczr2XnHO8iDOlHefdyq2tb2xu5bcLO7t7+wf24VFThbGk0KAhD2XbIwo4C6ChmebQjiQQ4XFoeeObdN56AKlYGNzpSQQ9QYYB8xkl2kh9u1jrdwXRIykSf4q7igm4x78aC/xp3y45ZWcGvErcjJRQhnrf/u4OQhoLCDTlRKmO60S6lxCpGeUwLXRjBRGhYzKEjqEBEaB6ySzLFJ8aZYD9UJoXaDxT/14kRCg1EZ7ZTD2q5Vkq/jvzxMLPiTLWRjBYsqP9y55JHcUaAjp348cc6xCn7eEBk0A1nxhCqGQmEKYjIgnVpuOCacpd7mWVNM/LbqV8dVspVa+zzvLoBBXRGXLRBaqiGqqjBqLoET2hF/RqPVtv1of1OV/NWdnNMVqA9fUDKzGq8A==</latexit>

T = TR

<latexit sha1_base64="6anX4vLinp2i2UXkif+dCyX0CJo=">AAACEXicbVDLSsNAFJ3UV62vqks3g0VwVRIpqAuh6MZllaYttKFMJjft0MkkzEyEEvoNLtzop7gTt36BX+LWaZuFbT0wcDjnXu6Z4yecKW3b31ZhbX1jc6u4XdrZ3ds/KB8etVScSgoujXksOz5RwJkAVzPNoZNIIJHPoe2P7qZ++wmkYrFo6nECXkQGgoWMEm0kt3nT7D/2yxW7as+AV4mTkwrK0eiXf3pBTNMIhKacKNV17ER7GZGaUQ6TUi9VkBA6IgPoGipIBMrLZmEn+MwoAQ5jaZ7QeKb+3chIpNQ48s1kRPRQLXtT8V/PjxYuZ8pEG0KwFEeHV17GRJJqEHSeJkw51jGe1oMDJoFqPjaEUMnMhzAdEkmoNiWWTFPOci+rpHVRdWrV64dapX6bd1ZEJ+gUnSMHXaI6ukcN5CKKGHpGr+jNerHerQ/rcz5asPKdY7QA6+sXA3KdzQ==</latexit>

EOSw

<latexit sha1_base64="oTxWFQduV47F1iZIeUu187mprZc=">AAACDXicbVDLSgMxFM3UV62vqks3wSK4KjMiqLuiG5ct2Ae0Q8lk7rShSWZIMkoZ+gUu3OinuBO3foNf4ta0nYVtPRA4nHMv9+QECWfauO63U1hb39jcKm6Xdnb39g/Kh0ctHaeKQpPGPFadgGjgTELTMMOhkyggIuDQDkZ3U7/9CEqzWD6YcQK+IAPJIkaJsVLjqV+uuFV3BrxKvJxUUI56v/zTC2OaCpCGcqJ113MT42dEGUY5TEq9VENC6IgMoGupJAK0n82CTvCZVUIcxco+afBM/buREaH1WAR2UhAz1MveVPzXC8TC5UzbaEMIl+KY6NrPmExSA5LO00QpxybG02pwyBRQw8eWEKqY/RCmQ6IINbbAkm3KW+5llbQuqt5l9aZxWand5p0V0Qk6RefIQ1eohu5RHTURRYCe0St6c16cd+fD+ZyPFpx85xgtwPn6BY+QnIY=</latexit>

No late-time entropy production

Baryon-to-entropy ratio is fixed here

Usual reheating withnB

s< 1.2 1018

Hf

1013GeV

1/2 µ

1016GeV

4

8Hf/Mpl

<latexit sha1_base64="MCrm/wyujAAV5Cz2I/wDqKqT/qw=">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</latexit>

😢w 1/3

<latexit sha1_base64="2qe8tKo6CcL2oELnNiq3ISl5wFA=">AAACFHicbVC7TsMwFHXKq5RXgZHFokJiKglUArYKFsYi0YfURpXj3LZWbSfYDqiK+hMMLPApbIiVnS9hxW0z0MKRLB2dc6/u8QlizrRx3S8nt7S8srqWXy9sbG5t7xR39xo6ShSFOo14pFoB0cCZhLphhkMrVkBEwKEZDK8nfvMBlGaRvDOjGHxB+pL1GCXGSq3HDod77+SsWyy5ZXcK/Jd4GSmhDLVu8bsTRjQRIA3lROu258bGT4kyjHIYFzqJhpjQIelD21JJBGg/neYd4yOrhLgXKfukwVP190ZKhNYjEdhJQcxAL3oT8V8vEHOXU22jDSBciGN6F37KZJwYkHSWppdwbCI8aQiHTAE1fGQJoYrZD2E6IIpQY3ss2Ka8xV7+ksZp2auUL28rpepV1lkeHaBDdIw8dI6q6AbVUB1RxNETekGvzrPz5rw7H7PRnJPt7KM5OJ8/QZqe/Q==</latexit>

Courtesy H.Oide

It’s too early to give up!

If , effective entropy “reduction” can occur. w > 1/3

<latexit sha1_base64="tPvOlJ2ykBZxnLlY2YOQUdK001M=">AAACEXicbVDLSsNAFJ3UV62vqks3g0VwVRMtqBspunFZwbSFNpTJ5KYdOpOEmYlSQr/BhRv9FHfi1i/wS9w6bbOw1QMDh3Pu5Z45fsKZ0rb9ZRWWlldW14rrpY3Nre2d8u5eU8WppODSmMey7RMFnEXgaqY5tBMJRPgcWv7wZuK3HkAqFkf3epSAJ0g/YiGjRBvJfbxyTs565YpdtafAf4mTkwrK0eiVv7tBTFMBkaacKNVx7ER7GZGaUQ7jUjdVkBA6JH3oGBoRAcrLpmHH+MgoAQ5jaV6k8VT9vZERodRI+GZSED1Qi95E/NfzxdzlTJloAwgW4ujwwstYlKQaIjpLE6Yc6xhP6sEBk0A1HxlCqGTmQ5gOiCRUmxJLpilnsZe/pHladWrVy7tapX6dd1ZEB+gQHSMHnaM6ukUN5CKKGHpCL+jVerberHfrYzZasPKdfTQH6/MHgf6dfw==</latexit>

Courtesy H.Oide

Axion-like particle: shift-symmetric pseudo scalar

It is rather strange to assume the potential-driven inflation with matter-like reheating epoch.

?-1.0 -0.5 0.5 1.0

0.5

1.0

1.5

2.0

Courtesy H.Oide



It is not strange to consider the k-inflation with kination reheating epoch and gravitational reheating.

L(X,) =K()X

M2pl

+X2

M4pl

, X = 1

2(@)2

<latexit sha1_base64="PH2q4DlbRS/+U7UXmPr+irLuVr8=">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</latexit>

X ' M

2pl

2K(), 3H2

plM2pl '

1

4K

2()

<latexit sha1_base64="XsiR/o4gc9brMaVVimhreAn5URk=">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</latexit>

(’99 Armendariz-Picon, Damour & Mukhanov)

(’93 Spokoiny, ’96 Joyce)

Courtesy H.Oide



It is not strange to consider the k-inflation with kination reheating epoch and gravitational reheating.

L(X,) =K()X

M2pl

+X2

M4pl

, X = 1

2(@)2

<latexit sha1_base64="PH2q4DlbRS/+U7UXmPr+irLuVr8=">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</latexit>

X ' M

2pl

2K(), 3H2

plM2pl '

1

4K

2()

<latexit sha1_base64="XsiR/o4gc9brMaVVimhreAn5URk=">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</latexit>

Benefit: ' N22

Hinf

Mpl

3/2

<latexit sha1_base64="4eybBk5GJf5jUeMF9E62lqokhHE=">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</latexit>

<8Hinf

Mpl) N < 104

Hinf

105Mpl

1/2

<latexit sha1_base64="UNTyvBgERkOIgxL4DxWcDbZ9uaY=">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</latexit>

(’99 Armendariz-Picon, Damour & Mukhanov)


Courtesy H.Oide

Reheating after k-inflation: K() ! const.

<latexit sha1_base64="5m2hug8T7Umip9lW4v3lMVT6R7Q=">AAACMnicbVC7SgNBFJ31bXxFLW0Go6BN2JWA2ok2gk0EowF3CbOTSXZwHsvMXSUs+QG/xsJG/0Q7sbW3dTZJYaIHBg7n3Muce+JUcAu+/+ZNTc/Mzs0vLJaWlldW18rrG9dWZ4ayBtVCm2ZMLBNcsQZwEKyZGkZkLNhNfHdW+Df3zFiu1RX0UhZJ0lW8wykBJ7XKOxd7YZrwfRwa3k2AGKMfcCgJJEbmVCsL/SpulSt+1R8A/yXBiFTQCPVW+Ttsa5pJpoAKYu1t4KcQ5cQAp4L1S2FmWUroHemyW0cVkcxG+eCaPt51Sht3tHFPAR6ovzdyIq3tydhNFjntpFeI/3qxHPs5ty5awtoTcaBzFOVcpRkwRYdpOpnAoHHRH25zwyiIniOEGu4OwjQhhlBwLZdcU8FkL3/J9UE1qFWPL2uVk9NRZwtoC22jPRSgQ3SCzlEdNRBFj+gJvaBX79l79z68z+HolDfa2URj8L5+AJJHqw4=</latexit>

= p =K

M2pl

X

<latexit sha1_base64="176sfJ37JwrtJlIFxeLUtY4g83E=">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</latexit>

w = 1

<latexit sha1_base64="vzakCqlvu9lEy69JGpEGf8x1QZo=">AAACD3icbVDLSgMxFL1TX7W+qi7dBIvgqsxIQV0IRTcuK9oHtEPJZNI2NMkMSUYpQz/BhRv9FHfi1k/wS9yatrOwrQcCh3Pu5Z6cIOZMG9f9dnIrq2vrG/nNwtb2zu5ecf+goaNEEVonEY9UK8CaciZp3TDDaStWFIuA02YwvJn4zUeqNIvkgxnF1Be4L1mPEWysdP905XWLJbfsToGWiZeREmSodYs/nTAiiaDSEI61bntubPwUK8MIp+NCJ9E0xmSI+7RtqcSCaj+dRh2jE6uEqBcp+6RBU/XvRoqF1iMR2EmBzUAvehPxXy8Qc5dTbaMNaLgQx/Qu/JTJODFUklmaXsKRidCkHBQyRYnhI0swUcx+CJEBVpgYW2HBNuUt9rJMGmdlr1K+vKuUqtdZZ3k4gmM4BQ/OoQq3UIM6EOjDM7zCm/PivDsfzudsNOdkO4cwB+frF5F8nQg=</latexit>

EOS / a6

<latexit sha1_base64="u9EJbK6Svv5iejCaZ+AwTNrs9dY=">AAACIXicbVC7TsMwFHV4lvIKdGSxqJBYqBJU8dgqWBiLRB9SEyrHcRqrdhzZDlIU9VsYWOBT2BAb4kNYcR8DbTmSpaNz7tU9PkHKqNKO82WtrK6tb2yWtsrbO7t7+/bBYVuJTGLSwoIJ2Q2QIowmpKWpZqSbSoJ4wEgnGN6O/c4TkYqK5EHnKfE5GiQ0ohhpI/XtiidjAb1UilQLiB6Ls4tR3646NWcCuEzcGamCGZp9+8cLBc44STRmSKme66TaL5DUFDMyKnuZIinCQzQgPUMTxInyi0n4ETwxSggjIc1LNJyofzcKxJXKeWAmOdKxWvTG4r9ewOcuF8pEi0m4EEdHV35BkzTTJMHTNFHGoCljXBcMqSRYs9wQhCU1H4I4RhJhbUotm6bcxV6WSfu85tZr1/f1auNm1lkJHIFjcApccAka4A40QQtgkINn8ArerBfr3fqwPqejK9ZspwLmYH3/AsFUo/M=</latexit>


Courtesy H.Oide



= p =K

M2pl

X


w = 1


EOS / a6


a

<latexit sha1_base64="IpQ8ZiwT1PrnQDiNHD0vL26aw+0=">AAACDXicbVDLSgMxFM3UV62vqks3wSK4KjNSUHdFNy5bsA9oh5LJ3GlDk8yQZIQy9AtcuNFPcSdu/Qa/xK1pOwvbeiBwOOde7skJEs60cd1vp7CxubW9U9wt7e0fHB6Vj0/aOk4VhRaNeay6AdHAmYSWYYZDN1FARMChE4zvZ37nCZRmsXw0kwR8QYaSRYwSY6UmGZQrbtWdA68TLycVlKMxKP/0w5imAqShnGjd89zE+BlRhlEO01I/1ZAQOiZD6FkqiQDtZ/OgU3xhlRBHsbJPGjxX/25kRGg9EYGdFMSM9Ko3E//1ArF0OdM22gjClTgmuvEzJpPUgKSLNFHKsYnxrBocMgXU8IklhCpmP4TpiChCjS2wZJvyVntZJ+2rqler3jZrlfpd3lkRnaFzdIk8dI3q6AE1UAtRBOgZvaI358V5dz6cz8Vowcl3TtESnK9fawqccA==</latexit>

<latexit sha1_base64="jtPJmHSllVEx/luT4L+lH27/CBc=">AAACEHicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoO6KblxWsA9oh5LJpJ3QJDMkGaEM/QUXbvRT3Ilb/8AvcWumnYVtPRA4nHMv9+QECWfauO63U1pb39jcKm9Xdnb39g+qh0dtHaeK0BaJeay6AdaUM0lbhhlOu4miWAScdoLxXe53nqjSLJaPZpJQX+CRZENGsMmlvoriQbXm1t0Z0CrxClKDAs1B9acfxiQVVBrCsdY9z02Mn2FlGOF0WumnmiaYjPGI9iyVWFDtZ7OsU3RmlRANY2WfNGim/t3IsNB6IgI7KbCJ9LKXi/96gVi4nGkbLaLhUhwzvPYzJpPUUEnmaYYpRyZGeTsoZIoSwyeWYKKY/RAiEVaYGNthxTblLfeyStoXde+yfvNwWWvcFp2V4QRO4Rw8uIIG3EMTWkAggmd4hTfnxXl3PpzP+WjJKXaOYQHO1y/2jZ3S</latexit>

inf = 3H2infM

2inf

<latexit sha1_base64="Wq/dADyooORCKxZsX0Iv3MJEnEU=">AAACRHicbVDLSsNAFJ34rPUVdelmsAiuSlKLj4VQdNONUME+oKlhMpm0Q2eSMDMRSuif+DUu3OgP+A/uRNyJkzYL23pg4HDuvcw5x4sZlcqy3o2l5ZXVtfXCRnFza3tn19zbb8koEZg0ccQi0fGQJIyGpKmoYqQTC4K4x0jbG95k8/YjEZJG4b0axaTHUT+kAcVIack1zxwxiFyHIzUQPKVhML6Cp7A+ozxU4O2c4Jolq2xNABeJnZMSyNFwzW/Hj3DCSagwQ1J2bStWvRQJRTEj46KTSBIjPER90tU0RJzIXjrJN4bHWvFhEAn9QgUn6t+LFHEpR9zTm5lLOT/LxH9nHp/5OZXa2oD4c3ZUcNHTueNEkRBP3QQJgyqCWaPQp4JgxUaaICyoDgTxAAmEle69qJuy53tZJK1K2a6WL++qpdp13lkBHIIjcAJscA5qoA4aoAkweALP4BW8GS/Gh/FpfE1Xl4z85gDMwPj5BVFssuc=</latexit>

inflation end

/ a6


reheating

/ a4

<latexit sha1_base64="MdiAANpQKcVV+l8PzlTCgIcg5tk=">AAACIXicbVDLSgMxFM3UV62v0S7dBIvgxjIjBXVXdOOygn1AZyyZTKYNzSRDkhGGod/iwo1+ijtxJ36IW9N2Frb1QOBwzr3ckxMkjCrtOF9WaW19Y3OrvF3Z2d3bP7APjzpKpBKTNhZMyF6AFGGUk7ammpFeIgmKA0a6wfh26nefiFRU8AedJcSP0ZDTiGKkjTSwq54cCeglUiRaQPSYnzcmA7vm1J0Z4CpxC1IDBVoD+8cLBU5jwjVmSKm+6yTaz5HUFDMyqXipIgnCYzQkfUM5iony81n4CTw1SggjIc3jGs7Uvxs5ipXK4sBMxkiP1LI3Ff/1gnjhcq5MtBEJl+Lo6MrPKU9STTiep4lSBk0Z07pgSCXBmmWGICyp+RDEIyQR1qbUimnKXe5llXQu6m6jfn3fqDVvis7K4BicgDPggkvQBHegBdoAgww8g1fwZr1Y79aH9TkfLVnFThUswPr+Bb4Ao/E=</latexit>


Courtesy H.Oide



= p =K

M2pl

X


w = 1


EOS / a6


a



inf = 3H2infM

2inf


inflation end

/ a6


reheating

/ a4


nB

s= 9.7 1011

µ

1016GeV

4

105

Treh

107GeV

1

<latexit sha1_base64="bf7cms68jTMNdhEHp/e2/mln3u4=">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</latexit>


Courtesy H.Oide

Proof of concept for successful reheating

a



inf = 3H2infM

2inf


inflation end

/ a6


reheating

/ a4


104m

2H

2inf

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(cf. ’19 Hashiba & Yokoyama)

/ a3

<latexit sha1_base64="jds5/8oLN96TzvOFTTdWeCiah/0=">AAACIXicbVC7TsMwFHV4lvIKdGSxqJBYqBKoBGwVLIxFog+pCZXjOI1VO45sBymK+i0MLPApbIgN8SGsuI+BthzJ0tE59+oenyBlVGnH+bJWVtfWNzZLW+Xtnd29ffvgsK1EJjFpYcGE7AZIEUYT0tJUM9JNJUE8YKQTDG/HfueJSEVF8qDzlPgcDRIaUYy0kfp2xZOxgF4qRaoFRI/F2cWob1edmjMBXCbujFTBDM2+/eOFAmecJBozpFTPdVLtF0hqihkZlb1MkRThIRqQnqEJ4kT5xST8CJ4YJYSRkOYlGk7UvxsF4krlPDCTHOlYLXpj8V8v4HOXC2WixSRciKOjK7+gSZppkuBpmihj0JQxrguGVBKsWW4IwpKaD0EcI4mwNqWWTVPuYi/LpH1ec+u16/t6tXEz66wEjsAxOAUuuAQNcAeaoAUwyMEzeAVv1ov1bn1Yn9PRFWu2UwFzsL5/AbxWo/A=</latexit>

m Hinf ' 1013GeV

<latexit sha1_base64="UZGoy2/RjbRLN5HaEu1o2BvxbYk=">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</latexit>

) Treh ' 107GeV

<latexit sha1_base64="E5WMtkghAOa2tb9a16PzgSiaaTg=">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</latexit>

nB

s= 9.7 1011

µ

1016GeV

4

105

Treh

107GeV

1

<latexit sha1_base64="bf7cms68jTMNdhEHp/e2/mln3u4=">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</latexit>

Courtesy H.Oide

Summary

Courtesy H.Oide

1. Gravitational Leptogenesis is a baryogenesis mechanism that takes advantage of the gravitational chiral anomaly.

2. The ghost mode can be pushed above the cutoff scale of the theory so that it is not catastrophe of the scenario.

3. The UV divergence can be renormalized but renormalization condition cannot be fixed without the knowledge of the UV physics.

4. Nevertheless resultant baryon asymmetry is small in general. With somehow exotic assumtions, i) the UV cutoff-dependent baryon asymmetry, and ii) kination with gravitational reheating, it can finally make it to explain the present BAU.

towards a robust estimate for gravitational leptogenesis...courtesy h.oide towards a robust estimate...

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