EFT of scalar-tensor gravity
with timelike scalar profile
1.
2.
3.
4.
5.
6.

Introduction
EFT on Minkowski bkgd
EFT on cosmological bkgd
BH with timelike scalar profile
EFT on arbitrary bkgd
Appx A. Generalized 2nd law
Summary
Appx B. de Sitter entropy boundShinji Mukohyama(YITP, Kyoto U)
Ref. arXiv: 2204.00228 w/ V.Yingcharoenrat(arXiv: 2111.08119 w/ K.Aoki, M.A.Gorji & K.Takahashi)
Also Arkani-Hamed, Cheng, Luty and Mukohyama 2004 (hep-th/0312099)
Mukohyama 2005 (hep-th/0502189)

Collaborators

K.Aoki

M.A.Gorji

K.Takahashi

V.Yingcharoenrat

Ref. arXiv: 2204.00228 w/ V.Yingcharoenrat
arXiv: 2111.08119 w/ K.Aoki, M.A.Gorji & K.Takahashi

EFT of scalar-tensor gravity
with timelike scalar profile

INTRODUCTION

1.

Introduction

2.

EFT on Minkowski bkgd

3.

EFT on cosmological bkgd

4.

BH with timelike scalar profile5.

EFT on arbitrary bkgd

6.

Summary

Why gravity beyond GR?
(GR : general relativity)

A motivation for IR modification• Gravity at long distances
Flattening galaxy rotation curves
extra gravity
Dimming supernovae
accelerating universe• Usual explanation: new forms of matter(DARK MATTER) and energy (DARKENERGY).

Dark component in the solar system?
Precession of perihelion
observed in 1800’s…

Sun
Mercurywhich people tried to
explain with a “darkplanet”, Vulcan,

Sun
Mercury

But the right answer wasn’t “dark planet”, it was“change gravity” from Newton to GR.

Why gravity beyond GR?
• Can we address mysteries in the universe?
Dark energy, dark matter, inflation, big-bang singularity,
cosmic magnetic field, etc.

Why gravity beyond GR?
• Can we address mysteries in the universe?
Dark energy, dark matter, inflation, big-bang singularity,
cosmic magnetic field, etc.
• Help constructing a theory of quantum gravity?
Superstring, Horava-Lifshitz, etc.

Why gravity beyond GR?
• Can we address mysteries in the universe?
Dark energy, dark matter, inflation, big-bang singularity,
cosmic magnetic field, etc.
• Help constructing a theory of quantum gravity?
Superstring, Horava-Lifshitz, etc.
• Do we really understand GR?
One of the best ways to understand something may be
to break (modify) it and then to reconstruct it.
•…

EFT of scalar-tensor gravity
with timelike scalar profile
1.

Introduction

2.

EFT on Minkowski bkgd

3.

EFT on cosmological bkgd

4.

BH with timelike scalar profile5.

EFT on arbitrary bkgd

6.

Summary

EFT ON MINKOWSKIBACKGROUNDMany gravity theories• 3 check points“What are the physical d.o.f. ?”“How do they interact ?”“What is the regime of validity ?”• If two (or more) theories give the sameanswers to the 3 questions above then they
are the same even if they look different.
→ Effective Field Theory (EFT)
as universal description

Scalar-tensor theories of gravity• Metric gmn + scalar field f• Jordan (1955), Brans & Dicke (1961), Bergmann(1968), Wagoner (1970), …
• Most general scalar-tensor theory of gravity with
2nd order covariant EOM: Horndeski (1974)• DHOST theories beyond Horndeski: Langlois &Noui (2016)• U-DHOST theories beyond DHOST: DeFelice,
Langlois, Mukohyama, Noui & Wang (2018)• All of them (and more) are universally described byan effective field theory (EFT)EFT of inflation / dark energy= EFT of scalar-tensor gravity
with timelike scalar profile• Time diffeo is broken by the scalarprofile but spatial diffeo is preserved.
• All terms that respect spatial diffeo must be
included in the EFT action.
• Derivative & perturbative expansions• Diffeo can be restored by introducing
Nambu-Goldstone boson

EFT of scalar-tensor gravity
with timelike scalar profile• Time diffeo is broken by the scalarprofile but spatial diffeo is preserved.
• All terms that respect spatial diffeo must be
included in the EFT action.
• Derivative & perturbative expansions• Diffeo can be restored by introducing
Nambu-Goldstone boson

Simplest : ghost condensationref. Arkani-Hamed, Cheng, Luty, Mukohyama 2004

Systematic construction of
EFT of ghost condensation
Arkani-Hamed, Cheng, Luty and Mukohyama 2004

Backgrounds characterized by mf  0 and timelike

Background metric is Minkowski.
2
1
2

2
Leff = LEH + M ( h00 − 2 ) − 2 K +  
M

2

ij
i
j
− 2 K +    Kij + i j + 
M

4

(

)(

(

)

)


Gauge choice: f (t , x ) = t.   f = 0(Unitary gauge) Residual symmetry: x → x (t , x )Write down most general action invariant under
this residual symmetry.
(

Action for : undo unitary gauge!)

Start with flat backgroundg mn =  mn + hmn

hmn =  mn + n  mUnder residual ih00 = 0, h0i =  0i , hij =  i j +  ji

Action invariant under i Beginning at quadratic order,2
since we are assuming flat
( h00 ) OK2
space is good background.
( h0i )
1
Kij = (  0 hij −  j h0i −  i h0 j )K 2 , K ij Kij OK2
1 2  2 ij
2
Leff = LEH + M ( h00 ) − 2 K − 2 K K ij +
M
M

4





Action invariant under i Beginning at quadratic order,2
since we are assuming flat
( h00 ) OK2
space is good background.
( h0i )
1
Kij = (  0 hij −  j h0i −  i h0 j )K 2 , K ij Kij OK2
1 2  2 ij
2
Leff = LEH + M ( h00 ) − 2 K − 2 K K ij +MM4

Action for 0 = 



h00 → h00 − 2 0
Kij → Kij +  i  j

2
1
2

2
Leff = LEH + M ( h00 − 2 ) − 2 K +  
M

2

ij
i
j
− 2 K +    Kij + i j + 
M

4

(

)(

(

)

)

E → rE−1

dt → r dt
dx → r

−1 / 2dx

Make
invariant

2
2

1

(


)
3
2
dtd
x

−
+

2

M
2





 → r 1/ 4
2


(


)
3

Leading nonlinear operator in infrared  dtd x~2M

has scaling dimension 1/4. (Barely) irrelevant

Good low-E effective theory
Robust prediction
e.g. Ghost inflation [Arkani-hamed, Creminelli, Mukohyama, Zaldarriaga 2004]

Higgs mechanism Ghost condensate
Arkani-Hamed, Cheng, Luty and Mukohyama 2004

Orderparameter mf

V (|  |)P ( (f )2 )f


Instability

Tachyon

−m 
2

2

Ghost

−f

2

Condensate V’=0, V’’>0

P’=0, P’’>0

BrokenGauge symmetry
symmetry
Force to be Gauge force
modified
New force
Yukawa typelaw

Time translational
symmetry
Gravity
Newton+Oscillation

EFT of ghost condensation =EFT of scalar-tensor gravity with timelike
scalar profile on Minkowski backgroundArkani-Hamed, Cheng, Luty and Mukohyama 2004

Backgrounds characterized by  mf  0 and timelike
Background metric is Minkowski.
2
1
2

2
Leff = LEH + M ( h00 − 2 ) − 2 K +  M2− 2 K ij + i  j Kij + i j + 
M

4

(

)(

(

)

)

EFT of scalar-tensor gravity
with timelike scalar profile
1.

Introduction

2.

EFT on Minkowski bkgd

3.

EFT on cosmological bkgd

4.

BH with timelike scalar profile5.

EFT on arbitrary bkgd

6.

Summary

EFT ON COSMOLOGICALBACKGROUND

Extension to FLRW background= EFT of inflation/dark energy
Creminelli, Luty, Nicolis, Senatore 2006
Cheung, Creminelli, Fitzpatrick, Kaplan, Senatore 2007

• Action invariant under xi → xi(t,x)• Ingredients𝑔𝜇𝜈 , 𝑔𝜇𝜈 , 𝑅𝜇𝜈𝜌𝜎 , ∇𝜇 , t & its derivatives• 1st derivative of t

 m t =  m0
• 2nd derivative of t

Unitary gauge action

derivative & perturbative expansions

NG boson• Undo unitary gauge• NG boson in decoupling (subhorizon) limit• Sound speed
𝑐𝑠 : speed of propagation for modes with 𝜔 ≫ 𝐻2𝑘22
𝜔 ≃ 𝑐𝑠 2 for 𝜋 ∼ 𝐴 𝑡𝑎

exp −𝑖∫ 𝜔𝑑𝑡 + 𝑖𝑘 ⋅ 𝑥ԦApplication: non-Gaussinity of
inflationary perturbation 𝜻 = −𝑯𝝅power spectrum

non-Gaussianity

2 types of 3-point interactions
𝑐𝑠2 → size of non-Gaussianity∝
𝑐3 → shape of non-Gaussianity

𝒄𝟑 = 𝟎𝜅2

Prototype𝜅 of the3equilateral shape𝟏
for small 𝒄𝟐𝒔𝟐𝒄𝒔

plots of 𝐵𝜁 (𝑘, 𝜅2 𝑘, 𝜅3 𝑘)/𝐵𝜁 (𝑘, 𝑘, 𝑘)
𝒄𝟑 = −𝟑. 𝟔

𝜅2

Prototype of the
𝜅3orthogonalshape𝒄𝟑 = −𝟒. 𝟑

𝜅2Linear combination
𝜅3of the two shapes

Summary so far• Ghost condensation universally describes all
scalar-tensor theories of gravity with timelike
scalar profile on Minkowski background.
• Extension of ghost condensation to FLRW
backgrounds results in the EFT of inflation/DE.
• These EFTs provide universal descriptions of all
scalar-tensor theories of gravity with timelike
scalar profile on each background, including
Horndeski theory, DHOST theory, U-DHOST
theory, and more.

EFT of scalar-tensor gravity
with timelike scalar profile
1.

Introduction

2.

EFT on Minkowski bkgd

3.

EFT on cosmological bkgd

4.

BH with timelike scalar profile5.

EFT on arbitrary bkgd

6.

Summary

BLACK HOLE WITH TIMELIKESCALAR PROFILE• Cosmology and black holes (BHs) play as importantroles in gravitational physics as blackbody radiationand hydrogen atoms did in quantum mechanics.
• In cosmology a time-dependent scalar field can act
as dark energy (DE), while BHs serve as probes ofstrong gravity. We then hope to probe the scalar field
DE by astrophysical BHs.
• This would require the scalar field profile to betimelike near BH. Otherwise, contours of the scalar
field would become ill-defined.
• Are there BH solution with timelike scalar profile?

Stealth solutions in k-essence
Mukohyama 2005

• Action in Einstein frame• EOMs• Stealth sol with X = X0, where P’(X0)=0• X = X0 (≠ 0)

defines geodesic congruence
defines Gaussian normal coord.

Stealth solutions in k-essence
Mukohyama 2005

• Any metric locally admits Gaussian normal coord.
• If P’(X) has a real root X0 then any vacuum GR sol
withlocally leads to a stealth sol.
• Schwarzshild metric admits a “globally” well-behaved
Gaussian normal coord.
(Lemeitre reference frame)2
g mn dx dx = −d +
m

n

2

rg dR

+ r 2 ( , R )d 2r ( , R) 2/3

3
r ( , R) =  rg ( R −  ) 2• Stealth Schwarzschildsolution with,
if P’(X) has a positive root X0
and if Leff is canceled by LbareStealth solutions with f = qt+y(r)• Schwarzschild in k-essence (Mukohyama 2005)• Schwarzschild-(A)dS in Horndeski theory (Babichev &Charmousis 2013, Kobayashi & Tanahashi 2014) Schwarzshild-(A)dS
in DHOST (Ben Achour & Liu 2019, Motohashi & Minamitsuji 2019)• Kerr-(A)dS in DHOST (Charmousis & Crisotomi & Gregory &Stergioulas 2019)• However, perturbations around most of those stealthsolutions are infinitely strongly coupled (de Rham & Zhang
2019) . This means the solutions cannot be trusted.
• Approximately stealth solution in ghost condensate
does not suffer from strong coupling (Mukohyama 2005).
Why?

Origin of strong coupling• EFT around stealth Minkowski sol. (= ghost
condensate) → universal dispersion relation
without the usual k2 term
• For  = O(1) (>0), EFT is weakly coupled all
the way up to ~M.
• If eom’s for perturbations are strictly 2nd order(as in DHOST) then  = 0 and the dispersion
relation loses dependence on k
→ strong coupling• [For w2=cs2k2, strong coupling @ E~]

Strong coupling scales• EFT of inflation/DE in decoupling limit• If cs2 ≃ const is not too small,can beignored. We further assume 0 < cs< 1.

()

A solution: scordatura
Motohashi & Mukohyama 2019

• Detuning of degeneracy condition recovers
and uplifts the strong couplingscale to ~. If the amount of detuning
is small enough then an apparent ghost is
heavy enough to be integrated out.
• Scordatura = weak and controlled detuningof degeneracy condition• Scordatura DHOST realizes
ghost condensation near stealthsolutions while it behaves as
DHOST away from them.
educalingo.com

Strong coupling scales• De Sitter limit = small cs2 limitforApproximately stealth BH in
ghost condensate Mukohyama 2005
• Two time scales: tBH << tGC ~ MPl2/M3• For tBH << t << tGC, a usual BH sol is a good
approximation → approximately stealth
Schwarzschild metric:2rdR
g mn dx m dxn = −d 2 + g+ r 2 ( , R )d 2r ( , R)3
r ( , R) =  rg ( R −  ) 2

E = − m pm

f = M 22/3

 m =  +  R

Exact sol in the absenceof higher derivative terms

Approximately stealth BH in ghost
condensate
Mukohyama 2005; Cheng, Luty, Mukohyama and Thaler 2006

• A tiny tadpole due to higher derivative terms iscanceled by extremely slow time-dependence.
• As a result,  = f starts accreting gradually.
• XTE J1118+480 (Mbh~7Msun,r~3Rsun,t~240Myr or 7
Gyr)
M<1012GeV much weaker than M<100GeV 9 M 2  3M 2 v 2/3 PlM bh = M bh 0  1 +
 
2 
4 M Pl  4 M bh 0  v : advanced null coordinate : coefficient of h.d. term







Summary of stealth BH withtimelike scalar profile• If we want to learn something about scalar field DE from BH then weneed to consider BH solutions with timelike scalar profile• Stealth solutions = backgrounds with GR metric and non-trivial scalarprofile → examples of BH solutions with timelike scalar profile• Many of them suffer from strong coupling problem, which is solved byscordatura (= controlled detuning of degeneracy condition)• DHOST/Horndeski do not include scordatura but U-DHOST does
(DeFelice, Mukohyama, Takahashi 2022) .
• EFT of ghost condensation already includes scordatura.
Arkani-Hamed, Cheng, Luty and Mukohyama 2004

• Approximately stealth solution in scordatura is stealth at astrophysicalscales and is free from the strong coupling problem• Scordatura also helps recover the generalized second lawEFT of scalar-tensor gravity
with timelike scalar profile
1.

Introduction

2.

EFT on Minkowski bkgd

3.

EFT on cosmological bkgd

4.

BH with timelike scalar profile5.

EFT on arbitrary bkgd

6.

Summary

EFT ON ARBITRARYBACKGROUND
arXiv: 2204.00228 w/ V.Yingcharoenrat• Cosmology and black holes (BHs) play as importantroles in gravitational physics as blackbody radiationand hydrogen atoms did in quantum mechanics.
• In cosmology a time-dependent scalar field can act
as dark energy (DE), while BHs serve as probes ofstrong gravity. We then hope to probe the scalar field
DE by astrophysical BHs.
• This would require the scalar field profile to betimelike near BH. Otherwise, contours of the scalar
field would become ill-defined.
• Are there BH solution with timelike scalar profile?

• Cosmology and black holes (BHs) play as importantroles in gravitational physics as blackbody radiationand hydrogen atoms did in quantum mechanics.
• In cosmology a time-dependent scalar field can act
as dark energy (DE), while BHs serve as probes ofstrong gravity. We then hope to probe the scalar field
DE by astrophysical BHs.
• This would require the scalar field profile to betimelike near BH. Otherwise, contours of the scalar
field would become ill-defined.
• Are there BH solution with timelike scalar profile?
• Can we develop EFT of scalar-tensor gravity onsuch BH background?

It is not straightforward…
• General action in the unitary gauge (f = )• Taylor expansion around the background• The whole action is invariant under 3d diffeo
but each term is not…
• Each coefficient is a function of (, xi) but
cannot be promoted to an arbitrary function.

Solution: consistency relations• The chain rule

relates xi-derivatives of an EFT coefficient to other
EFT coefficients, and leads to consistency relations.
• The consistency relations ensure the spatialdiffeo invariance.
• Taylor coefficients should satisfy the consistency
relations but are otherwise arbitrary.
• (No consistency relation for -derivatives.)

EFT action• EFT coefficients should satisfy the consistency
relations but are otherwise arbitrary• One can restore 4d diffeo by Stueckelberg trick• Easy to find dictionary between EFT coefficientsand theory parameters• Can be applied to BH with timelike scalar profile• Bridge between theories and observations

Applications to BHs withtimelike scalar profile• Background analysis for spherical BH[arXiv: 2204.00228 w/ V.Yingcharoenrat]• Odd-parity perturbation around spherical BH→ Generalized Regge-Wheeler equation[arXiv: 2208.02943 w/ K.Takahashi & V.Yingcharoenrat][see also arXiv: 2208.02823 by Khoury, Noumi, Trodden, Wong]→ Quasi-normal mode
[work in progress w/ K.Takahashi & K.Tomikawa & V.Yingcharoenrat]• Even-parity perturbation around spherical BH[work in progress w/ K.Takahashi & V.Yingcharoenrat]• Rotating BH• Dynamical BH• …

EFT of scalar-tensor gravity
with timelike scalar profile

SUMMARY

1.

Introduction

2.

EFT on Minkowski bkgd

3.

EFT on cosmological bkgd

4.

BH with timelike scalar profile5.

EFT on arbitrary bkgd

6.

Summary• There are at least three motivations to go beyond GR:(i) mysteries in the universe; (ii) quantum gravity;(iii) understanding GR itself.
• Ghost condensation is a universal description of scalartensor gravity with timelike scalar profile on Minkowski
background.
• Extension of ghost condensation to FLRW backgrounds
results in a universal description of fluctuations ofinflaton/dark energy (DE), called EFT of inflation/DE.
• If we want to learn something about scalar field DE from a
black hole (BH) then we need to consider BH solutions withtimelike scalar profile.

EFT of scalar-tensor gravity with timelike scalar profile• Time diffeo is broken by the scalar profile but spatial diffeo is preserved.
• All terms that respect spatial diffeo must be included in the EFT action.
• Derivative & perturbative expansions• Diffeo can be restored by introducing NG boson
EFT of scalar-tensor gravity
on Minkowski background= ghost condensation

EFT of scalar-tensor gravity
on cosmological background= Creminelli,
EFT ofinflation/darkenergyLuty, Nicolis, Senatore 2006

Arkani-Hamed, Cheng, Luty and Mukohyama, JHEP 0405:074,2004

Cheung, Creminelli, Fitzpatrick, Kaplan, Senatore 2007

EFT of scalar-tensor gravity
on arbitrary background= EFT of BH perturbations
arXiv: 2204.00228 w/ Vicharit Yingcharoenrat

Taylor expansion of the general action

Consistency relations

S is invariant under spatial diffeo but the background breaks it.

• There are at least three motivations to go beyond GR:(i) mysteries in the universe; (ii) quantum gravity;(iii) understanding GR itself.
• Ghost condensation is a universal description of scalartensor gravity with timelike scalar profile on Minkowski
background.
• Extension of ghost condensation to FLRW backgrounds
results in a universal description of fluctuations ofinflaton/dark energy (DE), called EFT of inflation/DE.
• If we want to learn something about scalar field DE from a
black hole (BH) then we need to consider BH solutions withtimelike scalar profile.
• EFT of scalar-tensor gravity with timelike scalar profile on
arbitrary background was developed. Consistency relations
among EFT coefficients ensure the spatial diffeo invariance.
Applicable to BHs with scalar field DE.
• Other applications?

Further extension of the web of EFTs
“The Effective Field Theory of Vector-Tensor Theories”
Katsuki Aoki, Mohammad Ali Gorji, Shinji Mukohyama, Kazufumi Takahashi, , JCAP 01 (2022) 01, 059 [arXiv: 2111.08119].

Residual symmetry in the unitary gaugec.f. Residual symmetry in unitary gauge
for scalar-tensor theories

leaving

The web of EFTs

invariantArkani-Hamed, Cheng, Luty and Mukohyama 2004

Cheng, Luty, Mukohyama and Thaler 2004

Thank you!

K.Aoki

M.A.Gorji

K.Takahashi

V.Yingcharoenrat

Ref. arXiv: 2204.00228 w/ V.Yingcharoenrat
arXiv: 2111.08119 w/ K.Aoki, M.A.Gorji & K.Takahashi

EFT of scalar-tensor gravity
with timelike scalar profile
1.

Introduction

2.

EFT on Minkowski bkgd

3.

EFT on cosmological bkgd

4.

BH with timelike scalar profile5.

EFT on arbitrary bkgd

6.

Summary

Appendix A

GENERALIZED 2ND LAW
Mukohyama 2009, 2010

Higgs mechanism Ghost condensate
Arkani-Hamed, Cheng, Luty and Mukohyama 2004

Orderparameter mf

V (|  |)P ( (f )2 )f


Instability

Tachyon

−m 
2

2

Ghost

−f

2

Condensate V’=0, V’’>0

P’=0, P’’>0

BrokenGauge symmetry
symmetry
Force to be Gauge force
modified
New force
Yukawa typelaw

Time translational
symmetry
Gravity
Newton+Oscillation

Bounds on symmetry breaking scale M
Arkani-Hamed, Cheng, Luty and Mukohyama and Wiseman 2007

0100GeV

1TeV

allowedruled out

Jeans Instability(sun)Twinkling from Lensing(CMB)
Supernova time-delay

ruled out
ruled out

So far, there is no conflict with experiments
and observations if M < 100GeV.

M

EFT of ghost condensation =EFT of scalar-tensor gravity with timelike
scalar profile on Minkowski backgroundArkani-Hamed, Cheng, Luty and Mukohyama 2004

Backgrounds characterized by  mf  0 and timelike
Background metric is Minkowski.
2
1
2

2
Leff = LEH + M ( h00 − 2 ) − 2 K +  M2− 2 K ij + i  j Kij + i j + 
M

4

(

)(

(

)

)

Holography and GSL• Do holographic dual descriptions always exist?
PROBABLY NO. e.g.) A de Sitter space is only metastable and a unitary holographic dual is not known.
• How about ghost condensate?
• Let’s look for violation of GSL in ghost condensate,
since violation of GSL would indicate absence ofholographic dual. (GSL is expected to be dual toordinary 2nd law.)• Three proposals: (i) semi-classical heat flow; (ii)analogue of Penrose process; (iii) negative energy.

Different limits of speed−2A,B

g A,Bmn = −um un + c ( gmn + umun )

um =

 mf
−(f )2

 mf
M2 0
•
preferred direction um.
• Different particles A and B may follow
geodesics of different metrics gAmn and gBmn.
• Lorentz breaking effects such as |cA,B2-1|
vanish in the limit M2→0 (M2: order parameter)
cA,B2=1+O(M2/MPl2).

Semi-classical heat flow
Dubovsky and Sibiryakov 2006

TbhA < TshellA < TshellB < TbhB
BH

Shell A

Shell BSame |Fshell→bh|
Same black hole

BH

dSshell/dt =(1/TshellB-1/TshellA)*|Fshell→bh| < 0
dSbh/dt = 0 ???

Semi-classical heat flow
Dubovsky and Sibiryakov 2006; Mukohyama 2009

TbhA < TshellA < TshellB < TbhB
BH

Shell A

Shell BSame |Fshell→bh|
Same black hole

BH

dSshell/dt =(1/TshellB-1/TshellA)*|Fshell→bh| < 0
dSbh/dt = 0 ???

2/M 2)
notviolated!
• |TbhB- TGSL
|
/
T
=
O(M
bhA
bh
Pl
• |Fshell→bh| / Tbh2 = O(M2/MPl2)• |dSshell/dt| / Tbh = O(M4/MPl4)• dSbh/dt due to accretion is much larger.
• Stot=Sshell+Sbh does increase!

Analogue of Penrose process
Elling, Foster, Jacobson, Wall 2007
horizon A
m’B=0, E’B, p’BE’A< 0m’A=0, E’A, p’A
Dr=rg*O(M2/MPl2)
DSbh< 0 ???
horizon B
mB, EB, pB
mA, EA, pA

Analogue of Penrose process
Elling, Foster, Jacobson, Wall 2007; Mukohyama 2010
horizon A
m’B=0, E’B, p’BE’A< 0m’A=0, E’A, p’A
Dr=rg*O(M2/MPl2)
DSbh< 0 ???
horizon B
mB, EB, pB
mA, EA, pA

GSL not violated!

• cA < cB
horizon A outside horizon B• EA+EB=E’A+E’B, pA+pB=p’A+p’B
• Test particle approx.
mA,B/MPl2 <<(rgDr)1/2
mA,B2/Mbh2<<M2/MPl2|E’A|/Mbh<<M2/MPl2• This process takes time scale ~ rg , at least.
• DMbh,acc/Mbh~M2/MPl2
DSbh > 0!

Negative energy
Arkani-Hamed, talk at PI 2006
P(X)UV (ghost) instability

X=-(df)2

IR (gradient) instability
Ghost condensate
No instability

NEC violated
NEC OK

It appears that Sbh can be decreased by sending excitation with P’<0.

Averaged NEC
Mukohyama 2010

ActionI =  dx

4

X = − f mf
m− g P( X )Stress-energy tensor

Tmn = (  + P )um un + Pg mn

 = 2P ' X − P

EOM & shift chargeum =J m = −2 P '  mf
In the regime of validity of EFT (||<<1)

Q =  d J m u

P = M  p0 + p2  + O (  ) 

 = MX − 1m Jm = 0
4

2

1
2

m

3

 mf
X

m

4

2
3

 + P − M J m u = M  p2  + O (  ) 
4

4

Averaged NEC d (  + P )  M Q
2

Q0
for
d

(

+
P
)

0


Negative energy
Arkani-Hamed, talk at PI 2006; Mukohyama 2010
P(X)UV (ghost) instability

X=-(df)2

IR (gradient) instability
Ghost condensate
No instability

NEC violated
NEC OKGSL not violated!

It appears that Sbh can be decreased by sending excitation with P’<0.

• GSL in a coarse-grained sense can be protected by
the averaged NEC if the shift charge is non-negative.
(Negative energy is followed by larger positiveenergy.)• Negative charge states are plugged by instabilities inthe early universe if the shift symmetry is exact. (|P’|would be large in the early universe.)

EFT of scalar-tensor gravity
with timelike scalar profile
1.

Introduction

2.

EFT on Minkowski bkgd

3.

EFT on cosmological bkgd

4.

BH with timelike scalar profile5.

EFT on arbitrary bkgd

6.

Summary

Appendix B

DE SITTER ENTROPY BOUND
Jazayeri, Mukohyama, Saitou, Watanabe 2016

Holography and GSL• Do holographic dual descriptions always exist?
PROBABLY NO. e.g.) A de Sitter space is only metastable and a unitary holographic dual is not known.
• How about ghost condensate?
• Let’s look for violation of GSL in ghost condensate,
since violation of GSL would indicate absence ofholographic dual. (GSL is expected to be dual toordinary 2nd law.)• Three proposals: (i) semi-classical heat flow; (ii)analogue of Penrose process; (iii) negative energy.
• The generalized 2nd law can hold in the presenceof ghost condensate. (Mukohyama 2009, 2010)

Ghost inflation and de Sitterentropy boundJazayeri, Mukohyama, Saitou, Watanabe 2016

• Black holes & cosmology in gravity theories are
as important as Hydrogen atoms & blackbody
radiation in quantum mechanics• Provide non-trivial tests for theories of gravity
e.g. black-hole entropy in string theory• Does the theory of ghost condensation passthose tests?
• Ghost condensation can be consistent with BH
thermodynamics (Mukohyama 2009, 2010)• How about de Sitter thermodynamics?

de Sitter thermodynamics• de Sitter (dS) spacetime is one of the threespacetimes with maximal symmetry• dS horizon has temperature TH = H/(2)• In quantum gravity, a dS space is probably
unstable (e.g. KKLT, Susskind, …). So, let’s
consider a dS space as a part of inflation• Friedmann equation →1st law with entropy S = A/(4GN) = /(GNH2)(This is in contrast with analogue gravity systems.)

de Sitter entropy boundArkani-Hamed, et.al. 2007

• Slow roll inflation (non-eternal)

for non-eternal inflation

de Sitter entropy boundArkani-Hamed, et.al. 2007

• Eternal inflation• Fluctuation generated during eternal epoch
would collapse to form BH → unobservable!
• This bound holds for a large class of models
of inflation• Does ghost inflation satisfy the bound?

Ghost inflation
Arkani-Hamed, Creminelli, Mukohyama and Zaldarriaga 2004Vf  0!

fScale-invariant perturbations
cf. tilted ghost inflation, Senatore (2004)

H

~ f

H
~ M 
 

Similar tohybrid inflation but
NOT SLOW ROLL

scaling dim of f ~ M 2

 ~ M  ( H / M )1/ 4
5/ 4[compareH
M Pl 

]Prediction of Large non-Gauss.
Leading non-linear interaction
1/4
H
non-G of ~   M 
1/5  
~     ( ) 2
M2

scaling dim of op.
2
2

1

(


)
3
2
dtd
x

−
+

2

M
2





[Really “0.1”  ( /  )1/ 5 ~ 10-2. VISIBLE.
In usual inflation, non-G ~ ( /  ) ~ 10-5 too small.]fNL ~ 82 -4/5, equilateral type
Planck 2015 constraint (equilateral type)

fNL

−4/5
−
0.6


 0.5
= -4±43 (68% CL statistical) ➔de Sitter entropy boundArkani-Hamed, et.al. 2007

• Eternal inflation• Fluctuation generated during eternal epoch
would collapse to form BH → unobservable!
• This bound holds for a large class of models
of inflation• Does ghost inflation satisfy the bound?
The answer appears to be “no” since Ntot canbe arbitrarily large. Swampland?

Lower bound on L?
Jazayeri, Mukohyama, Saitou, Watanabe 2016

• Tiny L prevents earlier inflationary modes
from being observed.
with• Nobs ~ ln(kmax/kmin)

S = /(GNH2)• In our universe, L=O(1) and thus thebound is well satisfied.

Cosmological Page time
Jazayeri, Mukohyama, Saitou, Watanabe 2016

• Hawking rad from BH → Srad = Sent increases
but SBH (≧ Sent) decreases → semi-classicaldescription should break down @ Page time,
when SBH ~ half of SBH,init• After inflation, we expect to see O(1) deviationfrom semi-classical description @ Page time,
when Nobs ~ Send
• For example, if L decays at a=adecay thenSummary of appendices• Ghost condensation is the simplest Higgs
phase of gravity.
• The low-E EFT is determined by the symmetry
breaking pattern. No ghost in the EFT.
• Gravity is modified in IR.
• Consistent with experiments and observations
if M < 100GeV.
• It appears easy but is actually difficult to violate
the generalized 2nd law by ghost condensate.
• Ghost inflation predicts large non-Gaussianitythat can be tested.
• de Sitter entropy bound appears to be violated
but is actually satisfied by ghost inflation.