Ghost Condensation and Horizon Entropycosmo/SW_2016/PPT/Mukohyama_GC.pdfArkani-Hamed, Cheng, Luty and Mukohyama, JHEP 0405:074,2004 42 2 1 2 00 2 L L M h K eff EH 2 M D SS ...

Ghost Condensation

and

Horizon Entropy

Ref. arXiv: 1602.06511 w/ S.Jazayeri, R.Saitou and Y.Watanabe

Shinji Mukohyama

(YITP, Kyoto)

Based also on other works with

N. Arkani-Hamed, H.-C. Cheng, P. Creminelli, T. Furukawa, K. Ichiki, K.

Izumi, M. Luty, N. Sugiyama, J. Thaler, T. Wiseman, M. Zaldarriaga

Can we change gravity in IR?

Change Theory?Massive gravity Fierz-Pauli 1939DGP model Dvali-Gabadadze-Porrati 2000

Change State?Higgs phase of gravityThe simplest: Ghost condensationArkani-Hamed, Cheng, Luty and Mukohyama, JHEP 0405:074,2004.

Ghost condensation

Suppose

2)( LSMgrav LLL L

gravity

SMEnergy

Vacuum

Any coupling instability

Arkani-Hamed, Cheng, Luty and

Mukohyama, JHEP 0405:074,2004

In analogy with Higgs potential, can this be stabilized?

424 )()( ML ?Clearly are higher dim ops. Really we

should consider

4)(

()(~)( 22 QPPL )

Naively no sensible EFT.

NOT THE CASE.

There is a sensible EFT + derivative expansion.

(Shift symmetry is assumed.)

For simplicity

2)( PL

Eq of motion

0 PClearly any constant is a solution!

Suppose constant + timelike.

Go to frame where

ctc i 0,

Solution for any c!

P

in flat background.

Look at small fluctuations

ct

2222224 )()()(2)( cPcPccPxdS

So for 0)(2)( 222 cPccP

0)( 2 cP

NORMAL sign kinetic terms.

STABLE stable stable

In this language, we can have a good EFT because

the background sits at some value c.

doesn’t sample entire function .

Taylor expansion of around

makes perfect sense.

2)( P

Small fluctuations controlled by small # of

parameters at low energies.

2)( P 22)( c

So far, possible backgrounds are labeled by

continuous parameter c.

Situation changes in presence of gravity, in

expanding universe.2 2 2 2

4

( )

( )

ds dt a t dx

S d x gP g

T

~ PPg

E.O.M.

0][ 3 Pat 0P as a

0 or 0)( 2 P(unstable ghost

background)

P

2M

Gravity (Hubble friction)

drives you here!

In this language, we can have a good EFT because

the background sits at the special value.

doesn’t sample entire function .

Taylor expansion of around

makes perfect sense.

2)( P

Small fluctuations controlled by small # of

parameters at low energies.

2)( P 42)( M

Attractor

Look at small perturbations

2424444 ))(()()( MPMPMMPxd

No spatial kinetic term for !

Other terms ()(~ 2 QP )do contain spatial kinetic terms but at least

22 )(

22

2

24 )(2

1

MxdS

Low

energy

22

2

23 )(2

1

MxdPositive definite Hamiltonian

STABLE!

2

42

M

k

tM 2

Of course there are higher terms in

effective theory.

2

2224 )(

2

1

MxdS

Because of k4, spatial fluctuations are less

suppressed than we are used to…

To analyze low-energy theory, we must do

proper analysis of scaling dimensions of

operators!

2

2

~)(

M

2

2223 )(

2

1

Mxdtd

4/1

2/1

1

r

dxrdx

dtrdt

rEE

Make

invariant

Leading operator in infrared

2

23

~)(

Mxdtd

scaling dimension 1/4. (Barely) irrelevant

Good low-E effective theory.

HEALTHY SECTOR IN ABSENCE

OF GRAVITY

What makes the ghost condensate special is that,

unlike other cosmological fluid, IT DOES NOT

DILUTE AS UNIVERSE EXPANDS.

Normally:

In deep future, flat or dS (maximal symmetry)

EMPTY OF FLUID

NOTE: Background breaks Lorentz invariance

spontaneously – prefered frame where is spatially

isotropic. No different from CMBR or any

cosmological fluid.

NOT HERE!2M even as universe expands.

)()(44 MPgMPT

Exactly that of c.c. !)( 4MP

dS solution!

0 0

Flat!

Present acceleration can be due to ghost condensate

even with =0 (in the symmetric phase). w=p/r=-1

exactly. BUT NOT A C.C.

PHYSICAL FLUID WITH PHYSICAL

EXCITATIONS.

P P

and timelike

Background metric is maximally

symmetric, either Minkowski or dS.

Systematic construction of

Low-energy effective theory

Backgrounds characterized by

0

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

224 21

00 22eff EHL L M h K

M

22ij i j

ij i jK KM

Gauge choice: .),( txt

(Unitary gauge)

Residual symmetry: ),( xtxx

Write down most general action invariant under

this residual symmetry.

( Action for : undo unitary gauge!)

Start with flat background hg

h

Under residuali

ijjiijii hhh ,,0 0000

0

Action invariant under i

2

00h

2

0ih2 , ij ijK K K

0 0 01

2ij ij j i i jK h h h

OK

OK

24 21 2

00 2 2

ij

eff EH ijL L M h K K KM M

Action for

ij ij i jK K 00 00 02h h 0

224 21

00 22eff EHL L M h K

M

22ij i j

ij i jK KM

Beginning at quadratic order,

since we are assuming flat

space is good background.

2 23 2

2

1 ( )

2dtd x

M

4/1

2/1

1

r

dxrdx

dtrdt

rEE

Make

invariant

Leading nonlinear operator in infrared

2

23

~)(

Mxdtd

has scaling dimension 1/4. (Barely) irrelevant

Good low-E effective theory

Robust prediction

Higgs mechanism Ghost condensate

Order

parameter

Instability Tachyon Ghost

Condensate V’=0, V’’>0 P’=0, P’’>0

Broken

symmetry

Gauge symmetry Time translational

symmetry

Force to be

modified

Gauge force Gravity

New force

law

Yukawa type Newton+Oscillation

2 2 2

2( )P

(| |)V

Arkani-Hamed, Cheng, Luty and Mukohyama 2004

Bounds on symmetry breaking scale M

M

Jeans Instability

(sun)

100GeV 1TeV

Supernova time-delay

ruled out

ruled out

ruled out

0

allowed

Twinkling from Lensing

(CMB)

So far, there is no conflict with experiments

and observations if M < 100GeV.

Arkani-Hamed, Cheng, Luty and Mukohyama and Wiseman, JHEP 0701:036,2007

Ghost condensation• 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.

Holography and GSL• Do holographic dual descriptions always exist?

PROBABLY NO. e.g.) A de Sitter space is only meta-

stable 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 of

holographic dual. (GSL is expected to be dual to

ordinary 2nd law.)

• Three proposals: (i) semi-classical heat flow; (ii)

analogue of Penrose process; (iii) negative energy.

• The generalized 2nd law holds in the presence of

ghost condensate. (Mukohyama 2009, 2010)

Ghost inflation and de Sitter

entropy bound

• 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 pass

those tests?

• Ghost condensation is known to be consistent

with BH thermodynamics (Mukohyama 2009, 2010)

• How about de Sitter thermodynamics?

S.Jazayeri, S.Mukohyama, R.Saitou, Y.Watanabe 2016

de Sitter thermodynamics

• de Sitter (dS) spacetime is one of the three

spacetimes 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 bound

• Slow roll inflation (non-eternal)

Arkani-Hamed, et.al. 2007

for non-eternal inflation


• 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?


!0

NOT SLOW ROLL

Scale-invariant perturbationscf. tilted ghost inflation, Senatore (2004)

r

r

H~ 4/1)/( MHM ~

4/5

M

H~

~ 2M

scaling dim of

[compare ]Pl

H

M

Similar to

hybrid inflation but

Arkani-Hamed, Creminelli, Mukohyama and Zaldarriaga, JHEP 0404:001,2004

Ghost inflation

V

Prediction of Large non-Gauss.

Leading non-linear interaction

non-G of ~H

M

1/4 scaling dim of op.

1/5

r

r

~

2

2

( )

M

2 23 2

2

1 ( )

2dtd x

M

[Really “0.1” ~ 10-2. VISIBLE.

In usual inflation, non-G ~ ~ 10-5 too small.]

5/1/ rr rr /

fNL ~ 82 -4/5, equilateral type

Planck 2015 constraint (equilateral type)

fNL = -4±43 (68% CL statistical) 4/50.6 0.5


• 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 can

be arbitrarily large. Swampland?


Lower bound on ?

• Tiny prevents earlier inflationary modes

from being observed.

• Nobs ~ ln(kmax/kmin) S = /(GNH2)

• In our universe, W=O(1) and thus the

bound is well satisfied.

with


Summary• 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. (Mukohyama 2009, 2010)

• Ghost inflation predicts large non-Gaussianitythat can be tested.

• de Sitter entropy bound appears to be violated but is actually satisfied by ghost inflation.

Higgs mechanism Ghost condensate

Order

parameter

Instability Tachyon Ghost

Condensate V’=0, V’’>0 P’=0, P’’>0

Broken

symmetry

Gauge symmetry Time translational

symmetry

Force to be

modified

Gauge force Gravity

New force

law

Yukawa type Newton+Oscillation

2 2 2

2( )P

(| |)V

Thank you

very much!

Arkani-Hamed, Cheng, Luty and Mukohyama 2004

Cosmological Page time

• Hawking rad from BH Srad = Sent increases

but SBH (≧ Sent) decreases semi-classical description should break down @ Page time,

when SBH ~ half of SBH,init

• After inflation, we expect to see O(1) deviation

from semi-classical description @ Page time,

when Nobs ~ Send

• For example, if decays at a=adecay then


Ghost Condensation and Horizon Entropycosmo/SW_2016/PPT/Mukohyama_GC.pdfArkani-Hamed, Cheng, Luty and Mukohyama, JHEP 0405:074,2004 42 2 1 2 00 2 L L M h K eff EH 2 M D SS ...

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