Gravity as local-Lorentz gauge theory

Kin-ya Oda (Osaka)Theory:

Shinya Matsuzaki (Jilin) Shota Miyawaki (Osaka)

Masatoshi Yamada (Heidelberg) also

Kengo Kikuchi (Kyoto→RIKEN)

Phenomenology: Ryusuke Jinno (DESY) Mio Kubota (Ochanomizu) Seong Chan Park (Yonsei)

arXiv: 1904.05699

“InvariantTheoreticalInterpretationofInteraction”

GaugetheoryearlierthanYang-Mills(1954)Hehadregrettedthatitcouldhavebeen

RyoyuUtiyama(1956)

✝1990R.I.P.

Keyquestion:•Whatarerightoff-shelldegreesoffreedomtodescribe(quantum)gravity(just)belowstringscale?

• Currentstringtheorycannotanswer.

• Onlyexperimentcangivehints:

• PrimordialGWproducedduringinflation.

• Quantumgravitonfluctuationduringinflation.

• Intriguingpoint:Realquantumgravitysignal.(Thoughfree.)

Plan1. LocalLorentzsymmetrymorefundamentalthangeneralcovariance

2. GravityfromhiddenlocalLorentzsymmetry?

3. Palatini(Einstein-Cartan)vsmetricinHiggsinflation

Spinorincurvedspace• Howtodefinespinor(matter)oncurvedspace(inouruniverse)atall?

• Itʼsnota(non-trivial)representationofdiffeo(=gen.coord.t.f.)norofGL(4):

• Butoflocal-LorentzSO(1,3): (x) ! 0(x0) = (x)

<latexit sha1_base64="lpvTQUZhc/vr0CfxN1KSU97pj8k=">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</latexit>

(x) ! 0(x) = bS(⇤(x)) (x)

<latexit sha1_base64="tPVjiKjjzANPe5EhR2dN2Y2tLvs=">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</latexit>

Backup:ConcreteLL

• Forinfinitesimal:

(x) ! 0(x) = bS(⇤(x)) (x)

<latexit sha1_base64="tPVjiKjjzANPe5EhR2dN2Y2tLvs=">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</latexit>

bS(⇤(x)) = b1 + 1

2b⌃ab✓ab(x) + · · ·

<latexit sha1_base64="DRuvRXDiDv1f0MEFAPIs5YU3WAo=">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</latexit>

b⌃ab =1

4

⇥b�a, b�b

⇤

<latexit sha1_base64="wJeiQ94z95BqSHN2+fUWqSZ86pA=">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</latexit>

�b�a, b�b

= 2⌘abb1

<latexit sha1_base64="f9UAQlMwSCLFSvxcwJg/bnYhyQY=">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</latexit>

⇤ab(x) = �ab + ✓ab(x)

<latexit sha1_base64="n53BeFphPUEQYEijRxca9Iftfzk=">AAAJlXicpZZLb9NAEICnpUBTHm3hABKXQFQEQqo2qBIIqVIFCHFAog/6kOoQre1NY9Wxjb1JW9z8Af4AB06AOCB+Bhe4gjj0JyCOReLCgdmJ3TaPei3hKMns7My3M97xrM3AdSLJ2N7Q8ImRk6dOjxbGzpw9d358YvLCSuQ3Q0ssW77rh2smj4TreGJZOtIVa0EoeMN0xaq5+UDNr7ZEGDm+90zuBKLS4BueU3MsLlFVnZg1nqCxzZ8bJo/bVcM0jZoXb7dnDVu4ktRKecuQdSH7rKoTJTbN6Cr2C+VEKEFyzfuTI0UwwAYfLGhCAwR4IFF2gUOEn3UoA4MAdRWIURei5NC8gDaMoW8TrQRacNRu4u8GjtYTrYdjxYzI28JVXPyG6FmEKfaDfWT77Av7xH6yv8eyYmKoWHbw3+z4iqA6/ury0h+tVwP/JdQPvTJjllCDuxSrg7EHpFFZWB3/1svX+0v3Fqfi6+wd+4Xxv2V77DNm4LV+Wx8WxOIbpB8fkYmZcxwLpHbunYfyFt2XBkXq4U7EqPehhRapJOiOZdlL2E7sVbzbWn5AO6kya9MOV0ir9rSGvjeghDoDLRyMvI6amxpeBC8ou0G09T5aRUM7jmTgqJdloGU2rdaT5dX/ypRTBfbHtttH2s1BGsxS0grtuhyQb3gwl82vJxVRpycg29bGdbvrJ0ad7s6aGGeT+kCag9IePuXxEQuVh46nKtLOoKXzeVgbZN1d4b28ozZ5mBZ2IJ5JPLTIw2sknUF1XJ5B7bXLw95BvUuErQxyt1UervI9WrNpf3KpY4rkCdVRgqTagoMThRM3u9o4eXVkFYd9cCaoUZHmdQxTwzBzMCwNw8rBsDUM/dMXUJdMJYfOWd2qUeIxaE1draYdIj1Ra8lppt45yr1vGP3Cyu3p8sz0zMJMae5+8vYxClfgGvbgMtyBOXgM87CMK7+Hr/ANvhcuFWYLDwuPOqbDQ4nPRei6Ck//AVc1Ao8=</latexit>

⇤T⌘⇤ = ⌘

<latexit sha1_base64="zDKdyNajG0TGUHkeQ4OY9xVQVLI=">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</latexit>

Backup:DiffeovsGL(4)

• GL(4):

V µ ! @x0µ

@x⌫V ⌫

<latexit sha1_base64="LKxRYojB8ZcjinGtnTsfBPiTDq0=">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</latexit>

• Diffeo:

V µ ! Mµ⌫V

⌫

<latexit sha1_base64="aOI/XKosn44lNZKggqG5HoaN3RY=">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</latexit>

Differenceresidesinwhetherornottoput @�Mµ⌫ = @⌫M

µ�

<latexit sha1_base64="vYUYy0EFheFOvibiOFtdmf3xIGA=">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</latexit>

(NoLiederivativehere)

Vierbein(vielbein,tetrad)• Necessarytodefinespinor(matter)atalloncurvedspace(ouruniverse).

• Morefundamentalbuildingblockthanmetric:

gµ⌫(x) = ⌘ab eaµ(x) e

b⌫(x)

<latexit sha1_base64="NONlSawdHQggOFSmylBfNZnpcsA=">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</latexit>

eaµ ⇠ @⇠a

@xµ

<latexit sha1_base64="MoGwUD9kuiCDKUTBVSTYocgpcU8=">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</latexit>

Picturefromweb

Roughly,

LocalLorentzsymmetry

•Roughlylocalrotationoftangentspacebasis.

eaµ ⇠ @⇠a

@xµ

<latexit sha1_base64="MoGwUD9kuiCDKUTBVSTYocgpcU8=">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</latexit>

eaµ(x) ! ⇤ab(x) e

bµ(x)

<latexit sha1_base64="sBEx0dwGTi/mMr8i0QSRUHzH3ro=">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</latexit>

Vielbein:

Local-Lorentzgaugefieldω• Covariantderivativeonlocal-Lorentzvector:

• Roughly,gravitationalconnectionisΓ=e∂e­ω.(Thesamet.f.asChristoffel)

• Variationw.r.t.ω〜w.r.t.Γ.→Palatini!

DµVa(x) = @µV

a(x) + !abµ(x)V

b(x)

<latexit sha1_base64="/p4fWwLeLWEr26K8/+/3GtNagBc=">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</latexit>

Plan1. LocalLorentzsymmetrymorefundamentalthangeneralcovariance

2. GravityfromhiddenlocalLorentzsymmetry?

3. Palatini(Einstein-Cartan)vsmetricinHiggsinflation

Ongoingwork

Tofirstsummarizethispart

•“Linearrealization”allowsonlythreetermsatUV.

•PurelyauxiliaryorhiddenlocalLorentzsymmetry?

Peculiarityofgravity•“Matter”inLLfundamentalrepresentation,vierbeine,isspacetimevector.

•S=∫[eeee+eeF],F=dω+ωω.• LowestisF,notF2.Nokinetictermuptothisorder.•Botheandωareauxiliaryfields.•Theotheracquireskinetictermafteroneisintegratedout.

(“∧”abbreviated)

CC EH

Whatis“right”weak-fieldlimit?

• Conventionalweakfieldlimit:eaμ→δaμ.

• Whatweassert:eaμ→0.

[Floreanini,Percacci1996]

(e.g.onflatspacetime)• Non-linearrealization:φ→v.

• Linearrealization:φ→0.

✴Consequence:InversematricesgμνandeaμabsentatΛUV.

✴Correspondstonotputting inHiggsLagrangian.

HiggsGravity

Also,topologychange:[Tseytlin1982;Horowitz1991]

m6

'2

<latexit sha1_base64="Gp9fP/znMWgIChUIgJoOf1gPtaQ=">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</latexit>

Only3termsallowedatΛUV• S=∫[eeee+eeF+ψDψ]

• Ingeneral,coefficientsarefunctionofscalars:φ,ψψ,etc.• Kinetictermonlyforspinors:CompositenessconditionatΛUV.

•→LLgaugesymmetryashiddenlocalsymmetry?Bando,Kugo,Yamawaki(1985,88)

CC EH D=d+ωF=dω+ωω

Kinetictermgeneration• OrdinaryscalarandgaugekinetictermsOK,justasinhiddenlocalsymmetry.

• HowaboutvierbeinandLL-gaugekineticterms?q

ω ω

q q

1PIω ω q

p

q

p+ q

ω ω= + + · · ·

= 0 + + · · ·Z��c[a�b]d(�µ�q2 � qµqµ) ��c[a�b]d(�µ�q2 � qµqµ) + · · ·

✴ Spinor1-loopcancelsout,somehowmiraculously,bothindirectcomputationandinFRG(Wilsonian,divergencefree,1-loopexact).(Preliminary)

✴ VierbeinandLLgaugefieldremainauxiliaryafterquantumcorrection?

Plan1. LocalLorentzsymmetrymorefundamentalthangeneralcovariance

2. GravityfromhiddenlocalLorentzsymmetry?

3. Palatini(Einstein-Cartan)vsmetricinHiggsinflation

Tofirstsummarizethispart

• WewillseethatthereisobservabledifferencebetweenmetricandPalatiniformulations,ifwegobeyondEHtruncation.

• Thesearedifferenttheoriesingeneral.

• Asaconcreteexample,welookatHiggsinflation.

Planck Collaboration: Constraints on Inflation

0.94 0.96 0.98 1.00

Primordial tilt (ns)

0.00

0.05

0.10

0.15

0.20

Ten

sor-

to-s

cala

rra

tio

(r0.0

02)

Convex

Concave

TT,TE,EE+lowE+lensing

TT,TE,EE+lowE+lensing+BK14

TT,TE,EE+lowE+lensing+BK14+BAO

Natural inflation

Hilltop quartic model

� attractors

Power-law inflation

R2 inflation

V � �2

V � �4/3

V � �

V � �2/3

Low scale SB SUSYN�=50

N�=60

Fig. 8. Marginalized joint 68 % and 95 % CL regions for ns and r at k = 0.002 Mpc�1 from Planck alone and in combination withBK14 or BK14 plus BAO data, compared to the theoretical predictions of selected inflationary models. Note that the marginalizedjoint 68 % and 95 % CL regions assume dns/d ln k = 0.

limits obtained from a ⇤CDM-plus-tensor fit. We refer the inter-ested reader to PCI15 for a concise description of the inflationarymodels studied here and we limit ourselves here to a summaryof the main results of this analysis.

– The inflationary predictions (Mukhanov & Chibisov 1981;Starobinsky 1983) originally computed for the R2 model(Starobinsky 1980) to lowest order,

ns � 1 ' �2N, r '

12N2 , (48)

are in good agreement with Planck 2018 data, confirm-ing the previous 2013 and 2015 results. The 95 % CL al-lowed range 49 < N⇤ < 58 is compatible with the R2 ba-sic predictions N⇤ = 54, corresponding to Treh ⇠ 109 GeV(Bezrukov & Gorbunov 2012). A higher reheating temper-ature Treh ⇠ 1013 GeV, as predicted in Higgs inflation(Bezrukov & Shaposhnikov 2008), is also compatible withthe Planck data.

– Monomial potentials (Linde 1983) V(�) = �M4Pl (�/MPl)p

with p � 2 are strongly disfavoured with respect to theR2 model. For these values the Bayesian evidence is worsethan in 2015 because of the smaller level of tensor modesallowed by BK14. Models with p = 1 or p = 2/3(Silverstein & Westphal 2008; McAllister et al. 2010, 2014)are more compatible with the data.

– There are several mechanisms which could lower the pre-dictions for the tensor-to-scalar ratio for a given potentialV(�) in single-field inflationary models. Important exam-ples are a subluminal inflaton speed of sound due to a non-standard kinetic term (Garriga & Mukhanov 1999), a non-minimal coupling to gravity (Spokoiny 1984; Lucchin et al.

1986; Salopek et al. 1989; Fakir & Unruh 1990), or an ad-ditional damping term for the inflaton due to dissipation inother degrees of freedom, as in warm inflation (Berera 1995;Bastero-Gil et al. 2016). In the following we report on theconstraints for a non-minimal coupling to gravity of the typeF(�)R with F(�) = M2

Pl + ⇠�2. To be more specific, a quartic

potential, which would be excluded at high statistical signif-icance for a minimally-coupled scalar inflaton as seen fromTable 5, can be reconciled with Planck and BK14 data for⇠ > 0: we obtain a 95 % CL lower limit log10 ⇠ > �1.6 withln B = �1.6.

– Natural inflation (Freese et al. 1990; Adams et al. 1993) isdisfavoured by the Planck 2018 plus BK14 data with a Bayesfactor ln B = �4.2.

– Within the class of hilltop inflationary models(Boubekeur & Lyth 2005) we find that a quartic poten-tial provides a better fit than a quadratic one. In the quarticcase we find the 95 % CL lower limit log10(µ2/MPl) > 1.1.

– D-brane inflationary models (Kachru et al. 2003; Dvali et al.2001; Garcıa-Bellido et al. 2002) provide a good fit toPlanck and BK14 data for a large portion of their parame-ter space.

– For the simple one parameter class of inflationary potentialswith exponential tails (Goncharov & Linde 1984; Stewart1995; Dvali & Tye 1999; Burgess et al. 2002; Cicoli et al.2009) we find ln B = �1.0.

– Planck 2018 data strongly disfavour the hybrid model drivenby logarithmic quantum corrections in spontaneously brokensupersymmetric (SUSY) theories (Dvali et al. 1994), withln B = �5.0.

18

2σ-allowedregion

Dead☠

Dying☠Dying☠

Dead☠

Dying☠

Higgs inflation best fits

Higgs inflation! (Metric)

Dying☠

Palatinivsmetric

• MetricvsPalatiniformalismsinEinsteingravity:• Metric:ConnectionΓpre-determinedbymetricgasChristoffelsymbols.• Palatini:VaryΓasindependentvariable,werecoversameEinsteineq.

• DoesmakedifferenttheoryforHiggsinflation.• CanseethedifferenceinCMBB-modes:LiteBIRD,POLARBEAR-2,andCORE.• Evenindirectcosmicgravitational-wavebackground:(Ultimate)DECIGO.

S =

Zgµ⌫Rµ⌫(�)

<latexit sha1_base64="DkZP0XhJC6OhP303Gd9OJBHzI6w=">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</latexit>

⌦(�)<latexit sha1_base64="m9aHjiaerEYW8PpJMl6L4pjm14Q=">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</latexit>

R(�) = d�+ ��<latexit sha1_base64="oj81aM5dkjCP+2SlSmT/y4Uqbxg=">AAAHrXicpZVLb9NAEICnLeBSHk3hgsQlJSpqhRTWgASqhFTBAY5tStJKcYjW9iax6he2E1rcnJH4Axw4gcQBIa7wA7jwBzj0JyCOReLCgdm1k7Z5eCPVUbyzszOfZ2Zfum9bYUTIwdT0zJmz55TZ83MXLl66PJ9buFIJvXZgsLLh2V6wrdOQ2ZbLypEV2WzbDxh1dJtt6TuP+fhWhwWh5bnPoj2f1RzadK2GZdAIVfXcYmlZe0Idh648zGtmI5FvJU3yrucKpEjEkx8W1FQoQPqsewszq6CBCR4Y0AYHGLgQoWwDhRB/VVCBgI+6GsSoC1CyxDiDLsyhbxutGFpQ1O7gu4m9aqp1sc+ZofA28Cs2/gP0zMMS+Uk+kUPyg3wmv8i/saxYMHgse9jqiS/z6/Nvrm3+lXo52EbQOvLKjDmCBjwQsVoYuy80PAsj8e+8enu4uVpaim+SD+Q3xv+eHJDvmIHb+WN83GCld0gfH5GOmVPsM6QmtXNRfinq4ohIXZyJGPUedNCiJzFRsSz7CHZTex7vrpTvi5nkmXXFDNeEls9pA32XoYA6DS0sjLyFmhUJL4QXIrtRtOoQrSahjSNp2BtkaWiZTWsMZLl4qkypWIHDse0PkfYnII1mcakiZj0akW/QH8vmt9IV0RI7INvWxO+eXD8x6mSV1THOtjgHejlw7dEuj49Z8DxkPL4izQxab3wSFvc6XtvePrLFzmbpSpJR/LQqfv/ko4KbXRUqvBKZx2H2zy7ey4txGUOXMPQJGIaEYUzAMCUM+SrxxW7uSZa4D7p4T6mDt9KwULlTVO8W1Y17hbVH6Y01C9fhBu5bFe7DGjyFdSjjN1/DF/gK35TbSlnRlOeJ6fRU6nMVTjxK8z+wBpmy</latexit>

Backup:NottoPalatini?

• Canrewritef(R)-gravityintoBrans-Dicke(w→0):

Iglesias,Kaloper,Padilla,Park(2006)

Ç 6B`bi- r2 K�F2 +H2�` r?�i ydy3XRRej +`BiB+Bx2bX 6QHHQrBM; i?2 �mi?Q`b- r2 +QMbB/2`;2M2`�H f(R) ;`�pBiv- r?B+? +�M #2 `2r`Bii2M rBi? �M �mtBHB�`v }2H/ � �b

S =

Zd4x

p�gf(R) ! S =

Zd4x

p�g [f(�) + f 0(�)(R� �)] . UyXRV

LQi2 i?�i MQ 7`�K2 Bbbm2 �`Bb2b bQ 7�`X h?Bb Bb w ! 0 HBKBi Q7 i?2 r2HH@FMQrM "H�Mb@.B+F2 i?2Q`v

S =

Zd4x

p�g

h�R� w

�gµ⌫rµ�r⌫�� V (�)

i, UyXkV

rBi? i?2 B/2MiB}+�iBQM f 0(�) = � �M/ V (�) = �f 0(�)� f(�)XLQr r2 +QMbB/2` i?2 �+iBQM UyXkV BM S�H�iBMB 7Q`KmH�iBQM- BX2X r2 i`2�i i?2 K2i`B+ gµ⌫�M/ +QMM2+iBQM �µ

⌫⇢ �b BM/2T2M/2MiX >2`2 b2p2`�H bi�M/TQBMib �`2 TQbbB#H2 r?2M ;BpBM;i?2 `2H�iBQM #2ir22M i?2b2 irQ [m�MiBiB2b- �b +�M #2 `2�/ Qz 7`QK a2+X k Q7 ydy3XRRejXq2 bmKK�`Bx2 i?2K ?2`2,

RX h?2 `2H�iBQM Bb /2`Bp2/ 7`QK i?2 KBMBKBx�iBQM Q7 i?2 �+iBQM rX`XiX i?2 +QMM2+iBQM�µ

⌫⇢X h?Bb +Q``2bTQM/b iQ 1[X U8V@UeV Q7 ydy3XRRejX h?2 `2H�iBQM Bb bvK#QHB+�HHv;Bp2M �b � ⇠ (� ·

p�gg)�1@(� ·

p�gg)X U1[mBp�H2Mi iQ Y = z/z Q7 ydy3XRRejXV

h?Bb Bb MQi?BM; #mi i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QK i?2 1BMbi2BM@7`�K2K2i`B+X ULQi2 i?�i �·p�gg ⇠

p�gEgE rBi? gE #2BM; i?2 1BMbi2BM@7`�K2 K2i`B+XV

kX h?2 `2H�iBQM Bb ;Bp2M #v BKTQbBM; � G�;`�M;2 KmHiBTHB2` BMpQHpBM; i?2 K2i`B+ �M/+QMM2+iBQMX A7 r2 �/QTi i?Bb pB2rTQBMi- UBM}MBi2HvV K�Mv TQbbB#BHBiB2b �`Bb2X aQK2Q7 i?2K �`2,URV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 1BMbi2BM@7`�K2 K2i`B+X h?Bb Bb 2[mBp�H2Mi iQ 1[X UdV Q7 ydy3XRRejX avK#QH@B+�HHv- KmHiBTHB2` = �(� � (

p�gEgE)�1@(

p�gEgE))X h?Bb Bb �(Y � z/z) BM i?2

H�M;m�;2 Q7 ydy3XRRej BM a2+X kXUkV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 K2i`B+ Q7 i?2 Q`B;BM�H 7`�K2 UyXkVX h?Bb Bb 2[mBp�H2Mi iQ 1[X UNV Q7 ydy3XRRej-Q` 1[XURNVX h?Bb Bb KmHiBTHB2` �(Y � x/x) BM i?2 H�M;m�;2 Q7 ydy3XRRej BM a2+X kX

�b 2tT2+i2/- RX ;Bp2b i?2 b�K2 /vM�KB+b �b kXURV- r?BH2 kXURV �M/ kXUkV ;Bp2 /Bz2`2Mi/vM�KB+b BM ;2M2`�HX h?2 QMHv 2t+2TiBQM Bb BM i?2 /2+QmTHBM; HBKBi w ! 1- BM r?B+?kXURV �M/ kXUkV ;Bp2 i?2 B/2MiB+�H T`2/B+iBQMX �HbQ- kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �bi?2 K2i`B+ 7Q`KmH�iBQM rBi? i?2 �+iBQM UyXkVX Uh?Bb Bb i`BpB�H- #2+�mb2 i?2 KmHiBTHB2`;m�`�Mi22b i?�i i?2 +QMM2+iBQM Bb G2pB@*BpBi� Q7 Q`B;BM�H@7`�K2 K2i`B+X h?Bb Bb MQi?BM;#mi i?2 K2i`B+ 7Q`KmH�iBQMXVLQr- r?�i Bb i?2 +`BiB+BbK Q7 ydy3XRRej\ h?2v `2[mB`2 i?�i i?2 +Q``2+i S�H�iBMB 7Q`@KmH�iBQM b?QmH/ ;Bp2 i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX U�b r2 rBHH b22b?Q`iHv- i?Bb Bb �+im�HHv `2�bQM�#H2 BM i?2B` +QMi2tiX a22 #2HQrXV h?Bb Bb b22M 2X;X 7`QKi?2 b2Mi2M+2 dzhQ ;2i +Q``2+i S�H�iBMB ;`�pBiB2b XXX r?B+? T`2b2`p2b i?2 +�MQMB+�H bi`m+@im`2 Q7 i?2 i?2Q`vǴ UkM/ b2Mi2M+2 BM 8X amKK�`v BM ydy3XRRejVX h?2v `2;�`/ i?Qb2b2imTb i?�i +?�M;2 i?2 ǵ+�MQMB+�H bi`m+im`2 Q7 i?2 i?2Q`vǶ UBX2X i?2 `2H�iBQMb r?B+??QH/ BM i?2 K2i`B+ 7Q`KmH�iBQMV �b mMDmbiB}2/X A7 bQ- pB2rTQBMib RX �M/ kXURV b?QmH/ #2/Bb+�`/2/- #2+�mb2 QMHv kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX

k

Ç 6B`bi- r2 K�F2 +H2�` r?�i ydy3XRRej +`BiB+Bx2bX 6QHHQrBM; i?2 �mi?Q`b- r2 +QMbB/2`;2M2`�H f(R) ;`�pBiv- r?B+? +�M #2 `2r`Bii2M rBi? �M �mtBHB�`v }2H/ � �b

S =

Zd4x

p�gf(R) ! S =

Zd4x

p�g [f(�) + f 0(�)(R� �)] . UyXRV

LQi2 i?�i MQ 7`�K2 Bbbm2 �`Bb2b bQ 7�`X h?Bb Bb w ! 0 HBKBi Q7 i?2 r2HH@FMQrM "H�Mb@.B+F2 i?2Q`v

S =

Zd4x

p�g

h�R� w

�gµ⌫rµ�r⌫�� V (�)

i, UyXkV

rBi? i?2 B/2MiB}+�iBQM f 0(�) = � �M/ V (�) = �f 0(�)� f(�)XLQr r2 +QMbB/2` i?2 �+iBQM UyXkV BM S�H�iBMB 7Q`KmH�iBQM- BX2X r2 i`2�i i?2 K2i`B+ gµ⌫�M/ +QMM2+iBQM �µ

⌫⇢ �b BM/2T2M/2MiX >2`2 b2p2`�H bi�M/TQBMib �`2 TQbbB#H2 r?2M ;BpBM;i?2 `2H�iBQM #2ir22M i?2b2 irQ [m�MiBiB2b- �b +�M #2 `2�/ Qz 7`QK a2+X k Q7 ydy3XRRejXq2 bmKK�`Bx2 i?2K ?2`2,

RX h?2 `2H�iBQM Bb /2`Bp2/ 7`QK i?2 KBMBKBx�iBQM Q7 i?2 �+iBQM rX`XiX i?2 +QMM2+iBQM�µ

⌫⇢X h?Bb +Q``2bTQM/b iQ 1[X U8V@UeV Q7 ydy3XRRejX h?2 `2H�iBQM Bb bvK#QHB+�HHv;Bp2M �b � ⇠ (� ·

p�gg)�1@(� ·

p�gg)X U1[mBp�H2Mi iQ Y = z/z Q7 ydy3XRRejXV

h?Bb Bb MQi?BM; #mi i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QK i?2 1BMbi2BM@7`�K2K2i`B+X ULQi2 i?�i �·p�gg ⇠

p�gEgE rBi? gE #2BM; i?2 1BMbi2BM@7`�K2 K2i`B+XV

kX h?2 `2H�iBQM Bb ;Bp2M #v BKTQbBM; � G�;`�M;2 KmHiBTHB2` BMpQHpBM; i?2 K2i`B+ �M/+QMM2+iBQMX A7 r2 �/QTi i?Bb pB2rTQBMi- UBM}MBi2HvV K�Mv TQbbB#BHBiB2b �`Bb2X aQK2Q7 i?2K �`2,URV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 1BMbi2BM@7`�K2 K2i`B+X h?Bb Bb 2[mBp�H2Mi iQ 1[X UdV Q7 ydy3XRRejX avK#QH@B+�HHv- KmHiBTHB2` = �(� � (

p�gEgE)�1@(

p�gEgE))X h?Bb Bb �(Y � z/z) BM i?2

H�M;m�;2 Q7 ydy3XRRej BM a2+X kXUkV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 K2i`B+ Q7 i?2 Q`B;BM�H 7`�K2 UyXkVX h?Bb Bb 2[mBp�H2Mi iQ 1[X UNV Q7 ydy3XRRej-Q` 1[XURNVX h?Bb Bb KmHiBTHB2` �(Y � x/x) BM i?2 H�M;m�;2 Q7 ydy3XRRej BM a2+X kX

�b 2tT2+i2/- RX ;Bp2b i?2 b�K2 /vM�KB+b �b kXURV- r?BH2 kXURV �M/ kXUkV ;Bp2 /Bz2`2Mi/vM�KB+b BM ;2M2`�HX h?2 QMHv 2t+2TiBQM Bb BM i?2 /2+QmTHBM; HBKBi w ! 1- BM r?B+?kXURV �M/ kXUkV ;Bp2 i?2 B/2MiB+�H T`2/B+iBQMX �HbQ- kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �bi?2 K2i`B+ 7Q`KmH�iBQM rBi? i?2 �+iBQM UyXkVX Uh?Bb Bb i`BpB�H- #2+�mb2 i?2 KmHiBTHB2`;m�`�Mi22b i?�i i?2 +QMM2+iBQM Bb G2pB@*BpBi� Q7 Q`B;BM�H@7`�K2 K2i`B+X h?Bb Bb MQi?BM;#mi i?2 K2i`B+ 7Q`KmH�iBQMXVLQr- r?�i Bb i?2 +`BiB+BbK Q7 ydy3XRRej\ h?2v `2[mB`2 i?�i i?2 +Q``2+i S�H�iBMB 7Q`@KmH�iBQM b?QmH/ ;Bp2 i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX U�b r2 rBHH b22b?Q`iHv- i?Bb Bb �+im�HHv `2�bQM�#H2 BM i?2B` +QMi2tiX a22 #2HQrXV h?Bb Bb b22M 2X;X 7`QKi?2 b2Mi2M+2 dzhQ ;2i +Q``2+i S�H�iBMB ;`�pBiB2b XXX r?B+? T`2b2`p2b i?2 +�MQMB+�H bi`m+@im`2 Q7 i?2 i?2Q`vǴ UkM/ b2Mi2M+2 BM 8X amKK�`v BM ydy3XRRejVX h?2v `2;�`/ i?Qb2b2imTb i?�i +?�M;2 i?2 ǵ+�MQMB+�H bi`m+im`2 Q7 i?2 i?2Q`vǶ UBX2X i?2 `2H�iBQMb r?B+??QH/ BM i?2 K2i`B+ 7Q`KmH�iBQMV �b mMDmbiB}2/X A7 bQ- pB2rTQBMib RX �M/ kXURV b?QmH/ #2/Bb+�`/2/- #2+�mb2 QMHv kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX

k

Ç 6B`bi- r2 K�F2 +H2�` r?�i ydy3XRRej +`BiB+Bx2bX 6QHHQrBM; i?2 �mi?Q`b- r2 +QMbB/2`;2M2`�H f(R) ;`�pBiv- r?B+? +�M #2 `2r`Bii2M rBi? �M �mtBHB�`v }2H/ � �b

S =

Zd4x

p�gf(R) ! S =

Zd4x

p�g [f(�) + f 0(�)(R� �)] . UyXRV

LQi2 i?�i MQ 7`�K2 Bbbm2 �`Bb2b bQ 7�`X h?Bb Bb w ! 0 HBKBi Q7 i?2 r2HH@FMQrM "H�Mb@.B+F2 i?2Q`v

S =

Zd4x

p�g

h�R� w

�gµ⌫rµ�r⌫�� V (�)

i, UyXkV

rBi? i?2 B/2MiB}+�iBQM f 0(�) = � �M/ V (�) = �f 0(�)� f(�)XLQr r2 +QMbB/2` i?2 �+iBQM UyXkV BM S�H�iBMB 7Q`KmH�iBQM- BX2X r2 i`2�i i?2 K2i`B+ gµ⌫�M/ +QMM2+iBQM �µ

⌫⇢ �b BM/2T2M/2MiX >2`2 b2p2`�H bi�M/TQBMib �`2 TQbbB#H2 r?2M ;BpBM;i?2 `2H�iBQM #2ir22M i?2b2 irQ [m�MiBiB2b- �b +�M #2 `2�/ Qz 7`QK a2+X k Q7 ydy3XRRejXq2 bmKK�`Bx2 i?2K ?2`2,

RX h?2 `2H�iBQM Bb /2`Bp2/ 7`QK i?2 KBMBKBx�iBQM Q7 i?2 �+iBQM rX`XiX i?2 +QMM2+iBQM�µ

⌫⇢X h?Bb +Q``2bTQM/b iQ 1[X U8V@UeV Q7 ydy3XRRejX h?2 `2H�iBQM Bb bvK#QHB+�HHv;Bp2M �b � ⇠ (� ·

p�gg)�1@(� ·

p�gg)X U1[mBp�H2Mi iQ Y = z/z Q7 ydy3XRRejXV

h?Bb Bb MQi?BM; #mi i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QK i?2 1BMbi2BM@7`�K2K2i`B+X ULQi2 i?�i �·p�gg ⇠

p�gEgE rBi? gE #2BM; i?2 1BMbi2BM@7`�K2 K2i`B+XV

kX h?2 `2H�iBQM Bb ;Bp2M #v BKTQbBM; � G�;`�M;2 KmHiBTHB2` BMpQHpBM; i?2 K2i`B+ �M/+QMM2+iBQMX A7 r2 �/QTi i?Bb pB2rTQBMi- UBM}MBi2HvV K�Mv TQbbB#BHBiB2b �`Bb2X aQK2Q7 i?2K �`2,URV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 1BMbi2BM@7`�K2 K2i`B+X h?Bb Bb 2[mBp�H2Mi iQ 1[X UdV Q7 ydy3XRRejX avK#QH@B+�HHv- KmHiBTHB2` = �(� � (

p�gEgE)�1@(

p�gEgE))X h?Bb Bb �(Y � z/z) BM i?2

H�M;m�;2 Q7 ydy3XRRej BM a2+X kXUkV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 K2i`B+ Q7 i?2 Q`B;BM�H 7`�K2 UyXkVX h?Bb Bb 2[mBp�H2Mi iQ 1[X UNV Q7 ydy3XRRej-Q` 1[XURNVX h?Bb Bb KmHiBTHB2` �(Y � x/x) BM i?2 H�M;m�;2 Q7 ydy3XRRej BM a2+X kX

�b 2tT2+i2/- RX ;Bp2b i?2 b�K2 /vM�KB+b �b kXURV- r?BH2 kXURV �M/ kXUkV ;Bp2 /Bz2`2Mi/vM�KB+b BM ;2M2`�HX h?2 QMHv 2t+2TiBQM Bb BM i?2 /2+QmTHBM; HBKBi w ! 1- BM r?B+?kXURV �M/ kXUkV ;Bp2 i?2 B/2MiB+�H T`2/B+iBQMX �HbQ- kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �bi?2 K2i`B+ 7Q`KmH�iBQM rBi? i?2 �+iBQM UyXkVX Uh?Bb Bb i`BpB�H- #2+�mb2 i?2 KmHiBTHB2`;m�`�Mi22b i?�i i?2 +QMM2+iBQM Bb G2pB@*BpBi� Q7 Q`B;BM�H@7`�K2 K2i`B+X h?Bb Bb MQi?BM;#mi i?2 K2i`B+ 7Q`KmH�iBQMXVLQr- r?�i Bb i?2 +`BiB+BbK Q7 ydy3XRRej\ h?2v `2[mB`2 i?�i i?2 +Q``2+i S�H�iBMB 7Q`@KmH�iBQM b?QmH/ ;Bp2 i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX U�b r2 rBHH b22b?Q`iHv- i?Bb Bb �+im�HHv `2�bQM�#H2 BM i?2B` +QMi2tiX a22 #2HQrXV h?Bb Bb b22M 2X;X 7`QKi?2 b2Mi2M+2 dzhQ ;2i +Q``2+i S�H�iBMB ;`�pBiB2b XXX r?B+? T`2b2`p2b i?2 +�MQMB+�H bi`m+@im`2 Q7 i?2 i?2Q`vǴ UkM/ b2Mi2M+2 BM 8X amKK�`v BM ydy3XRRejVX h?2v `2;�`/ i?Qb2b2imTb i?�i +?�M;2 i?2 ǵ+�MQMB+�H bi`m+im`2 Q7 i?2 i?2Q`vǶ UBX2X i?2 `2H�iBQMb r?B+??QH/ BM i?2 K2i`B+ 7Q`KmH�iBQMV �b mMDmbiB}2/X A7 bQ- pB2rTQBMib RX �M/ kXURV b?QmH/ #2/Bb+�`/2/- #2+�mb2 QMHv kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX

k

Ç 6B`bi- r2 K�F2 +H2�` r?�i ydy3XRRej +`BiB+Bx2bX 6QHHQrBM; i?2 �mi?Q`b- r2 +QMbB/2`;2M2`�H f(R) ;`�pBiv- r?B+? +�M #2 `2r`Bii2M rBi? �M �mtBHB�`v }2H/ � �b

S =

Zd4x

p�gf(R) ! S =

Zd4x

p�g [f(�) + f 0(�)(R� �)] . UyXRV

LQi2 i?�i MQ 7`�K2 Bbbm2 �`Bb2b bQ 7�`X h?Bb Bb w ! 0 HBKBi Q7 i?2 r2HH@FMQrM "H�Mb@.B+F2 i?2Q`v

S =

Zd4x

p�g

h�R� w

�gµ⌫rµ�r⌫�� V (�)

i, UyXkV

rBi? i?2 B/2MiB}+�iBQM f 0(�) = � �M/ V (�) = �f 0(�)� f(�)XLQr r2 +QMbB/2` i?2 �+iBQM UyXkV BM S�H�iBMB 7Q`KmH�iBQM- BX2X r2 i`2�i i?2 K2i`B+ gµ⌫�M/ +QMM2+iBQM �µ

⌫⇢ �b BM/2T2M/2MiX >2`2 b2p2`�H bi�M/TQBMib �`2 TQbbB#H2 r?2M ;BpBM;i?2 `2H�iBQM #2ir22M i?2b2 irQ [m�MiBiB2b- �b +�M #2 `2�/ Qz 7`QK a2+X k Q7 ydy3XRRejXq2 bmKK�`Bx2 i?2K ?2`2,

RX h?2 `2H�iBQM Bb /2`Bp2/ 7`QK i?2 KBMBKBx�iBQM Q7 i?2 �+iBQM rX`XiX i?2 +QMM2+iBQM�µ

⌫⇢X h?Bb +Q``2bTQM/b iQ 1[X U8V@UeV Q7 ydy3XRRejX h?2 `2H�iBQM Bb bvK#QHB+�HHv;Bp2M �b � ⇠ (� ·

p�gg)�1@(� ·

p�gg)X U1[mBp�H2Mi iQ Y = z/z Q7 ydy3XRRejXV

h?Bb Bb MQi?BM; #mi i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QK i?2 1BMbi2BM@7`�K2K2i`B+X ULQi2 i?�i �·p�gg ⇠

p�gEgE rBi? gE #2BM; i?2 1BMbi2BM@7`�K2 K2i`B+XV

kX h?2 `2H�iBQM Bb ;Bp2M #v BKTQbBM; � G�;`�M;2 KmHiBTHB2` BMpQHpBM; i?2 K2i`B+ �M/+QMM2+iBQMX A7 r2 �/QTi i?Bb pB2rTQBMi- UBM}MBi2HvV K�Mv TQbbB#BHBiB2b �`Bb2X aQK2Q7 i?2K �`2,URV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 1BMbi2BM@7`�K2 K2i`B+X h?Bb Bb 2[mBp�H2Mi iQ 1[X UdV Q7 ydy3XRRejX avK#QH@B+�HHv- KmHiBTHB2` = �(� � (

p�gEgE)�1@(

p�gEgE))X h?Bb Bb �(Y � z/z) BM i?2

H�M;m�;2 Q7 ydy3XRRej BM a2+X kXUkV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 K2i`B+ Q7 i?2 Q`B;BM�H 7`�K2 UyXkVX h?Bb Bb 2[mBp�H2Mi iQ 1[X UNV Q7 ydy3XRRej-Q` 1[XURNVX h?Bb Bb KmHiBTHB2` �(Y � x/x) BM i?2 H�M;m�;2 Q7 ydy3XRRej BM a2+X kX

�b 2tT2+i2/- RX ;Bp2b i?2 b�K2 /vM�KB+b �b kXURV- r?BH2 kXURV �M/ kXUkV ;Bp2 /Bz2`2Mi/vM�KB+b BM ;2M2`�HX h?2 QMHv 2t+2TiBQM Bb BM i?2 /2+QmTHBM; HBKBi w ! 1- BM r?B+?kXURV �M/ kXUkV ;Bp2 i?2 B/2MiB+�H T`2/B+iBQMX �HbQ- kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �bi?2 K2i`B+ 7Q`KmH�iBQM rBi? i?2 �+iBQM UyXkVX Uh?Bb Bb i`BpB�H- #2+�mb2 i?2 KmHiBTHB2`;m�`�Mi22b i?�i i?2 +QMM2+iBQM Bb G2pB@*BpBi� Q7 Q`B;BM�H@7`�K2 K2i`B+X h?Bb Bb MQi?BM;#mi i?2 K2i`B+ 7Q`KmH�iBQMXVLQr- r?�i Bb i?2 +`BiB+BbK Q7 ydy3XRRej\ h?2v `2[mB`2 i?�i i?2 +Q``2+i S�H�iBMB 7Q`@KmH�iBQM b?QmH/ ;Bp2 i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX U�b r2 rBHH b22b?Q`iHv- i?Bb Bb �+im�HHv `2�bQM�#H2 BM i?2B` +QMi2tiX a22 #2HQrXV h?Bb Bb b22M 2X;X 7`QKi?2 b2Mi2M+2 dzhQ ;2i +Q``2+i S�H�iBMB ;`�pBiB2b XXX r?B+? T`2b2`p2b i?2 +�MQMB+�H bi`m+@im`2 Q7 i?2 i?2Q`vǴ UkM/ b2Mi2M+2 BM 8X amKK�`v BM ydy3XRRejVX h?2v `2;�`/ i?Qb2b2imTb i?�i +?�M;2 i?2 ǵ+�MQMB+�H bi`m+im`2 Q7 i?2 i?2Q`vǶ UBX2X i?2 `2H�iBQMb r?B+??QH/ BM i?2 K2i`B+ 7Q`KmH�iBQMV �b mMDmbiB}2/X A7 bQ- pB2rTQBMib RX �M/ kXURV b?QmH/ #2/Bb+�`/2/- #2+�mb2 QMHv kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX

k

Ç 6B`bi- r2 K�F2 +H2�` r?�i ydy3XRRej +`BiB+Bx2bX 6QHHQrBM; i?2 �mi?Q`b- r2 +QMbB/2`;2M2`�H f(R) ;`�pBiv- r?B+? +�M #2 `2r`Bii2M rBi? �M �mtBHB�`v }2H/ � �b

S =

Zd4x

p�gf(R) ! S =

Zd4x

p�g [f(�) + f 0(�)(R� �)] . UyXRV

LQi2 i?�i MQ 7`�K2 Bbbm2 �`Bb2b bQ 7�`X h?Bb Bb w ! 0 HBKBi Q7 i?2 r2HH@FMQrM "H�Mb@.B+F2 i?2Q`v

S =

Zd4x

p�g

h�R� w

�gµ⌫rµ�r⌫�� V (�)

i, UyXkV

rBi? i?2 B/2MiB}+�iBQM f 0(�) = � �M/ V (�) = �f 0(�)� f(�)XLQr r2 +QMbB/2` i?2 �+iBQM UyXkV BM S�H�iBMB 7Q`KmH�iBQM- BX2X r2 i`2�i i?2 K2i`B+ gµ⌫�M/ +QMM2+iBQM �µ

⌫⇢ �b BM/2T2M/2MiX >2`2 b2p2`�H bi�M/TQBMib �`2 TQbbB#H2 r?2M ;BpBM;i?2 `2H�iBQM #2ir22M i?2b2 irQ [m�MiBiB2b- �b +�M #2 `2�/ Qz 7`QK a2+X k Q7 ydy3XRRejXq2 bmKK�`Bx2 i?2K ?2`2,

RX h?2 `2H�iBQM Bb /2`Bp2/ 7`QK i?2 KBMBKBx�iBQM Q7 i?2 �+iBQM rX`XiX i?2 +QMM2+iBQM�µ

⌫⇢X h?Bb +Q``2bTQM/b iQ 1[X U8V@UeV Q7 ydy3XRRejX h?2 `2H�iBQM Bb bvK#QHB+�HHv;Bp2M �b � ⇠ (� ·

p�gg)�1@(� ·

p�gg)X U1[mBp�H2Mi iQ Y = z/z Q7 ydy3XRRejXV

h?Bb Bb MQi?BM; #mi i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QK i?2 1BMbi2BM@7`�K2K2i`B+X ULQi2 i?�i �·p�gg ⇠

p�gEgE rBi? gE #2BM; i?2 1BMbi2BM@7`�K2 K2i`B+XV

kX h?2 `2H�iBQM Bb ;Bp2M #v BKTQbBM; � G�;`�M;2 KmHiBTHB2` BMpQHpBM; i?2 K2i`B+ �M/+QMM2+iBQMX A7 r2 �/QTi i?Bb pB2rTQBMi- UBM}MBi2HvV K�Mv TQbbB#BHBiB2b �`Bb2X aQK2Q7 i?2K �`2,URV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 1BMbi2BM@7`�K2 K2i`B+X h?Bb Bb 2[mBp�H2Mi iQ 1[X UdV Q7 ydy3XRRejX avK#QH@B+�HHv- KmHiBTHB2` = �(� � (

p�gEgE)�1@(

p�gEgE))X h?Bb Bb �(Y � z/z) BM i?2

H�M;m�;2 Q7 ydy3XRRej BM a2+X kXUkV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 K2i`B+ Q7 i?2 Q`B;BM�H 7`�K2 UyXkVX h?Bb Bb 2[mBp�H2Mi iQ 1[X UNV Q7 ydy3XRRej-Q` 1[XURNVX h?Bb Bb KmHiBTHB2` �(Y � x/x) BM i?2 H�M;m�;2 Q7 ydy3XRRej BM a2+X kX

�b 2tT2+i2/- RX ;Bp2b i?2 b�K2 /vM�KB+b �b kXURV- r?BH2 kXURV �M/ kXUkV ;Bp2 /Bz2`2Mi/vM�KB+b BM ;2M2`�HX h?2 QMHv 2t+2TiBQM Bb BM i?2 /2+QmTHBM; HBKBi w ! 1- BM r?B+?kXURV �M/ kXUkV ;Bp2 i?2 B/2MiB+�H T`2/B+iBQMX �HbQ- kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �bi?2 K2i`B+ 7Q`KmH�iBQM rBi? i?2 �+iBQM UyXkVX Uh?Bb Bb i`BpB�H- #2+�mb2 i?2 KmHiBTHB2`;m�`�Mi22b i?�i i?2 +QMM2+iBQM Bb G2pB@*BpBi� Q7 Q`B;BM�H@7`�K2 K2i`B+X h?Bb Bb MQi?BM;#mi i?2 K2i`B+ 7Q`KmH�iBQMXVLQr- r?�i Bb i?2 +`BiB+BbK Q7 ydy3XRRej\ h?2v `2[mB`2 i?�i i?2 +Q``2+i S�H�iBMB 7Q`@KmH�iBQM b?QmH/ ;Bp2 i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX U�b r2 rBHH b22b?Q`iHv- i?Bb Bb �+im�HHv `2�bQM�#H2 BM i?2B` +QMi2tiX a22 #2HQrXV h?Bb Bb b22M 2X;X 7`QKi?2 b2Mi2M+2 dzhQ ;2i +Q``2+i S�H�iBMB ;`�pBiB2b XXX r?B+? T`2b2`p2b i?2 +�MQMB+�H bi`m+@im`2 Q7 i?2 i?2Q`vǴ UkM/ b2Mi2M+2 BM 8X amKK�`v BM ydy3XRRejVX h?2v `2;�`/ i?Qb2b2imTb i?�i +?�M;2 i?2 ǵ+�MQMB+�H bi`m+im`2 Q7 i?2 i?2Q`vǶ UBX2X i?2 `2H�iBQMb r?B+??QH/ BM i?2 K2i`B+ 7Q`KmH�iBQMV �b mMDmbiB}2/X A7 bQ- pB2rTQBMib RX �M/ kXURV b?QmH/ #2/Bb+�`/2/- #2+�mb2 QMHv kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX

k

Backup:NottoPalatini?Iglesias,Kaloper,Padilla,Park(2006)

Ç 6B`bi- r2 K�F2 +H2�` r?�i ydy3XRRej +`BiB+Bx2bX 6QHHQrBM; i?2 �mi?Q`b- r2 +QMbB/2`;2M2`�H f(R) ;`�pBiv- r?B+? +�M #2 `2r`Bii2M rBi? �M �mtBHB�`v }2H/ � �b

S =

Zd4x

p�gf(R) ! S =

Zd4x

p�g [f(�) + f 0(�)(R� �)] . UyXRV

LQi2 i?�i MQ 7`�K2 Bbbm2 �`Bb2b bQ 7�`X h?Bb Bb w ! 0 HBKBi Q7 i?2 r2HH@FMQrM "H�Mb@.B+F2 i?2Q`v

S =

Zd4x

p�g

h�R� w

�gµ⌫rµ�r⌫�� V (�)

i, UyXkV

rBi? i?2 B/2MiB}+�iBQM f 0(�) = � �M/ V (�) = �f 0(�)� f(�)XLQr r2 +QMbB/2` i?2 �+iBQM UyXkV BM S�H�iBMB 7Q`KmH�iBQM- BX2X r2 i`2�i i?2 K2i`B+ gµ⌫�M/ +QMM2+iBQM �µ

⌫⇢ �b BM/2T2M/2MiX >2`2 b2p2`�H bi�M/TQBMib �`2 TQbbB#H2 r?2M ;BpBM;i?2 `2H�iBQM #2ir22M i?2b2 irQ [m�MiBiB2b- �b +�M #2 `2�/ Qz 7`QK a2+X k Q7 ydy3XRRejXq2 bmKK�`Bx2 i?2K ?2`2,

RX h?2 `2H�iBQM Bb /2`Bp2/ 7`QK i?2 KBMBKBx�iBQM Q7 i?2 �+iBQM rX`XiX i?2 +QMM2+iBQM�µ

⌫⇢X h?Bb +Q``2bTQM/b iQ 1[X U8V@UeV Q7 ydy3XRRejX h?2 `2H�iBQM Bb bvK#QHB+�HHv;Bp2M �b � ⇠ (� ·

p�gg)�1@(� ·

p�gg)X U1[mBp�H2Mi iQ Y = z/z Q7 ydy3XRRejXV

h?Bb Bb MQi?BM; #mi i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QK i?2 1BMbi2BM@7`�K2K2i`B+X ULQi2 i?�i �·p�gg ⇠

p�gEgE rBi? gE #2BM; i?2 1BMbi2BM@7`�K2 K2i`B+XV

kX h?2 `2H�iBQM Bb ;Bp2M #v BKTQbBM; � G�;`�M;2 KmHiBTHB2` BMpQHpBM; i?2 K2i`B+ �M/+QMM2+iBQMX A7 r2 �/QTi i?Bb pB2rTQBMi- UBM}MBi2HvV K�Mv TQbbB#BHBiB2b �`Bb2X aQK2Q7 i?2K �`2,URV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 1BMbi2BM@7`�K2 K2i`B+X h?Bb Bb 2[mBp�H2Mi iQ 1[X UdV Q7 ydy3XRRejX avK#QH@B+�HHv- KmHiBTHB2` = �(� � (

p�gEgE)�1@(

p�gEgE))X h?Bb Bb �(Y � z/z) BM i?2

H�M;m�;2 Q7 ydy3XRRej BM a2+X kXUkV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 K2i`B+ Q7 i?2 Q`B;BM�H 7`�K2 UyXkVX h?Bb Bb 2[mBp�H2Mi iQ 1[X UNV Q7 ydy3XRRej-Q` 1[XURNVX h?Bb Bb KmHiBTHB2` �(Y � x/x) BM i?2 H�M;m�;2 Q7 ydy3XRRej BM a2+X kX

�b 2tT2+i2/- RX ;Bp2b i?2 b�K2 /vM�KB+b �b kXURV- r?BH2 kXURV �M/ kXUkV ;Bp2 /Bz2`2Mi/vM�KB+b BM ;2M2`�HX h?2 QMHv 2t+2TiBQM Bb BM i?2 /2+QmTHBM; HBKBi w ! 1- BM r?B+?kXURV �M/ kXUkV ;Bp2 i?2 B/2MiB+�H T`2/B+iBQMX �HbQ- kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �bi?2 K2i`B+ 7Q`KmH�iBQM rBi? i?2 �+iBQM UyXkVX Uh?Bb Bb i`BpB�H- #2+�mb2 i?2 KmHiBTHB2`;m�`�Mi22b i?�i i?2 +QMM2+iBQM Bb G2pB@*BpBi� Q7 Q`B;BM�H@7`�K2 K2i`B+X h?Bb Bb MQi?BM;#mi i?2 K2i`B+ 7Q`KmH�iBQMXVLQr- r?�i Bb i?2 +`BiB+BbK Q7 ydy3XRRej\ h?2v `2[mB`2 i?�i i?2 +Q``2+i S�H�iBMB 7Q`@KmH�iBQM b?QmH/ ;Bp2 i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX U�b r2 rBHH b22b?Q`iHv- i?Bb Bb �+im�HHv `2�bQM�#H2 BM i?2B` +QMi2tiX a22 #2HQrXV h?Bb Bb b22M 2X;X 7`QKi?2 b2Mi2M+2 dzhQ ;2i +Q``2+i S�H�iBMB ;`�pBiB2b XXX r?B+? T`2b2`p2b i?2 +�MQMB+�H bi`m+@im`2 Q7 i?2 i?2Q`vǴ UkM/ b2Mi2M+2 BM 8X amKK�`v BM ydy3XRRejVX h?2v `2;�`/ i?Qb2b2imTb i?�i +?�M;2 i?2 ǵ+�MQMB+�H bi`m+im`2 Q7 i?2 i?2Q`vǶ UBX2X i?2 `2H�iBQMb r?B+??QH/ BM i?2 K2i`B+ 7Q`KmH�iBQMV �b mMDmbiB}2/X A7 bQ- pB2rTQBMib RX �M/ kXURV b?QmH/ #2/Bb+�`/2/- #2+�mb2 QMHv kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX

k

• Theirclaim:

（PuttingLagrangemultiplier,）ImposemetricityΓ=Γ(g)onJordan-framemetric.（＝Allowonlymetrictheoryfromthebeginning.）

• Reasoning:1. Otherwise,onegetsill-defined(stronglycoupled)w→0limit.(“Otherwise”=IfoneputsmultiplierforEinstein-framemetric.)

2. Otherwise,therewouldappearinfiniteambiguityofonwhatframeoneputsthemultiplier.

Backup:NottoPalatini?Iglesias,Kaloper,Padilla,Park(2006)

Ç 6B`bi- r2 K�F2 +H2�` r?�i ydy3XRRej +`BiB+Bx2bX 6QHHQrBM; i?2 �mi?Q`b- r2 +QMbB/2`;2M2`�H f(R) ;`�pBiv- r?B+? +�M #2 `2r`Bii2M rBi? �M �mtBHB�`v }2H/ � �b

S =

Zd4x

p�gf(R) ! S =

Zd4x

p�g [f(�) + f 0(�)(R� �)] . UyXRV

LQi2 i?�i MQ 7`�K2 Bbbm2 �`Bb2b bQ 7�`X h?Bb Bb w ! 0 HBKBi Q7 i?2 r2HH@FMQrM "H�Mb@.B+F2 i?2Q`v

S =

Zd4x

p�g

h�R� w

�gµ⌫rµ�r⌫�� V (�)

i, UyXkV

rBi? i?2 B/2MiB}+�iBQM f 0(�) = � �M/ V (�) = �f 0(�)� f(�)XLQr r2 +QMbB/2` i?2 �+iBQM UyXkV BM S�H�iBMB 7Q`KmH�iBQM- BX2X r2 i`2�i i?2 K2i`B+ gµ⌫�M/ +QMM2+iBQM �µ

⌫⇢ �b BM/2T2M/2MiX >2`2 b2p2`�H bi�M/TQBMib �`2 TQbbB#H2 r?2M ;BpBM;i?2 `2H�iBQM #2ir22M i?2b2 irQ [m�MiBiB2b- �b +�M #2 `2�/ Qz 7`QK a2+X k Q7 ydy3XRRejXq2 bmKK�`Bx2 i?2K ?2`2,

RX h?2 `2H�iBQM Bb /2`Bp2/ 7`QK i?2 KBMBKBx�iBQM Q7 i?2 �+iBQM rX`XiX i?2 +QMM2+iBQM�µ

⌫⇢X h?Bb +Q``2bTQM/b iQ 1[X U8V@UeV Q7 ydy3XRRejX h?2 `2H�iBQM Bb bvK#QHB+�HHv;Bp2M �b � ⇠ (� ·

p�gg)�1@(� ·

p�gg)X U1[mBp�H2Mi iQ Y = z/z Q7 ydy3XRRejXV

h?Bb Bb MQi?BM; #mi i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QK i?2 1BMbi2BM@7`�K2K2i`B+X ULQi2 i?�i �·p�gg ⇠

p�gEgE rBi? gE #2BM; i?2 1BMbi2BM@7`�K2 K2i`B+XV

kX h?2 `2H�iBQM Bb ;Bp2M #v BKTQbBM; � G�;`�M;2 KmHiBTHB2` BMpQHpBM; i?2 K2i`B+ �M/+QMM2+iBQMX A7 r2 �/QTi i?Bb pB2rTQBMi- UBM}MBi2HvV K�Mv TQbbB#BHBiB2b �`Bb2X aQK2Q7 i?2K �`2,URV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 1BMbi2BM@7`�K2 K2i`B+X h?Bb Bb 2[mBp�H2Mi iQ 1[X UdV Q7 ydy3XRRejX avK#QH@B+�HHv- KmHiBTHB2` = �(� � (

p�gEgE)�1@(

p�gEgE))X h?Bb Bb �(Y � z/z) BM i?2

H�M;m�;2 Q7 ydy3XRRej BM a2+X kXUkV h?2 KmHiBTHB2` Bb BKTQb2/ bQ i?�i � Bb i?2 G2pB@*BpBi� +QMM2+iBQM K�/2 7`QKi?2 K2i`B+ Q7 i?2 Q`B;BM�H 7`�K2 UyXkVX h?Bb Bb 2[mBp�H2Mi iQ 1[X UNV Q7 ydy3XRRej-Q` 1[XURNVX h?Bb Bb KmHiBTHB2` �(Y � x/x) BM i?2 H�M;m�;2 Q7 ydy3XRRej BM a2+X kX

�b 2tT2+i2/- RX ;Bp2b i?2 b�K2 /vM�KB+b �b kXURV- r?BH2 kXURV �M/ kXUkV ;Bp2 /Bz2`2Mi/vM�KB+b BM ;2M2`�HX h?2 QMHv 2t+2TiBQM Bb BM i?2 /2+QmTHBM; HBKBi w ! 1- BM r?B+?kXURV �M/ kXUkV ;Bp2 i?2 B/2MiB+�H T`2/B+iBQMX �HbQ- kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �bi?2 K2i`B+ 7Q`KmH�iBQM rBi? i?2 �+iBQM UyXkVX Uh?Bb Bb i`BpB�H- #2+�mb2 i?2 KmHiBTHB2`;m�`�Mi22b i?�i i?2 +QMM2+iBQM Bb G2pB@*BpBi� Q7 Q`B;BM�H@7`�K2 K2i`B+X h?Bb Bb MQi?BM;#mi i?2 K2i`B+ 7Q`KmH�iBQMXVLQr- r?�i Bb i?2 +`BiB+BbK Q7 ydy3XRRej\ h?2v `2[mB`2 i?�i i?2 +Q``2+i S�H�iBMB 7Q`@KmH�iBQM b?QmH/ ;Bp2 i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX U�b r2 rBHH b22b?Q`iHv- i?Bb Bb �+im�HHv `2�bQM�#H2 BM i?2B` +QMi2tiX a22 #2HQrXV h?Bb Bb b22M 2X;X 7`QKi?2 b2Mi2M+2 dzhQ ;2i +Q``2+i S�H�iBMB ;`�pBiB2b XXX r?B+? T`2b2`p2b i?2 +�MQMB+�H bi`m+@im`2 Q7 i?2 i?2Q`vǴ UkM/ b2Mi2M+2 BM 8X amKK�`v BM ydy3XRRejVX h?2v `2;�`/ i?Qb2b2imTb i?�i +?�M;2 i?2 ǵ+�MQMB+�H bi`m+im`2 Q7 i?2 i?2Q`vǶ UBX2X i?2 `2H�iBQMb r?B+??QH/ BM i?2 K2i`B+ 7Q`KmH�iBQMV �b mMDmbiB}2/X A7 bQ- pB2rTQBMib RX �M/ kXURV b?QmH/ #2/Bb+�`/2/- #2+�mb2 QMHv kXUkV ;Bp2b i?2 b�K2 T`2/B+iBQM �b i?2 K2i`B+ 7Q`KmH�iBQMX

k

• WhythisargumentdoesNOTapplytoHiggsinflation:

1. InHiggsinflation,scalardoeshavekineticterminJordan-frameaction.Therearenostrongcouplingproblem.(Now→0limitfromthebeginning.)

• Furthermore,Higgs-inflationactioncannotberewrittenintoBrans-Dickeformatall.

2. Varyingw.r.t.ΓuniquelycorrespondstoputtingmultiplierforEinstein-framemetric.

RobustnessofHiggsinflationprediction

• Conclusion:

• Palatinipredictiondeviatesbytinyinjectionofξ4〜10­5orλ6〜10­5/ξ2.(Nextslide)

• Theoreticallyunnaturalξ2≫1drivesmetrictheoryontoattractor.

• Natureseemstofavorthistheoreticallyunnatural

Jinno,Kubota,KO,Park(2019)

S =

Zd4x

p�g

R

2

�1 + ⇠2�

2 + ⇠4�4 + · · ·

�

+��0 + �4�

4 + �6�6 + · · ·

� �

<latexit sha1_base64="Xowid1/vE0zDtI6hrigqJo8dXLk=">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</latexit>

InfluenceonHiggsinflationlog10λ4

-14

-12

-10

-8

-6

-4

-2

0

ξ2ξ4

Figure 1: E↵ects of the higher dimensional operator ⇠4�4RJ in the metric (left) and Palatini (right)

formalisms with N = 60 (top) and 50 (bottom). We plot contours of fixed ⇠2 (blue, horizontal) andof fixed ⇠4 (red, vertical) in the ns-r plane. The value of �4 is also shown as a density plot. Allowedregions of 1� and 2� from the Planck experiment [9] (TT,TE,EE+lowE+lensing+BK14+BAO) arealso shown in the center.

9

Jinno,Kubota,KO,Park(2019)

Summary&discussion1. LocalLorentzsymmetrymorefundamentalthangeneralcovariance

2. GravityfromhiddenlocalLorentzsymmetry?

• Linearrealizationallowsonly3terms.

• PurelyauxiliaryPalatini,indicated.(Preliminary)

3. Palatini(Einstein-Cartan)vsmetricinHiggsinflation

• Natureseemstofavortheoreticallyunnatural

• Onpoint2:

• Anysymmetry(orwhatsoever)behind,tokeepeandωauxiliary?

• Todo:

• FullFRGanalysis,morethankinetictermsofeandω

• InclusionofΩ(φ)etc.

Thank you!