Living with the Dark En-ergy in Horava Gravity

Mu-In ParkKunsan Nat’al Univ., Korea

Based on arXiv:0905.4480 [JHEP], arXiv:0906.4275 [J-CAP],

APCTP-IEU Focus Program on Cosmology and Fun-damental Physics 2 (8 June 2011)

5) Quantum Gravity at a Lifshitz Point.Petr Horava, (UC, Berkeley & LBL, Berkeley) . Jan 2009. (Published Jan 2009). 29pp. Published in Phys.Rev.D79:084008,2009. e-Print: arXiv:0901.3775 [hep-th] TOPCITE = 250+ Cited 474 times

6) Membranes at Quantum Criticality.Petr Horava, (UC, Berkeley & LBL, Berkeley) . Dec 2008. 35pp. Published in JHEP 0903:020,2009. e-Print: arXiv:0812.4287 [hep-th] TOPCITE = 100+ Cited 258 times

1) General Covariance in Quantum Gravity at a Lifshitz Point.Petr Horava , Charles M. Melby-Thompson. Jul 2010. (Published Sep 15, 2010). 41pp.

Published in Phys.Rev.D82:064027,2010. e-Print: arXiv:1007.2410 [hep-th] Cited 50 times

APCTP Joint Focus Program:Frontiers of Black Hole Physics

December 6 ~ 17, 2010

• Topics -Horava-Lifshitz gravity

-Lorentz invariance violation-Loop quantum gravity-Acoustic black holes and experimenta-tion-Higher dimensional black holes (Kerr-Nut-(A)dS black holes)-Strings/ Branes-Other issues

We hear that GR is still good in IR.

• Confirmation of general relativity on large scales from weak lensing and galaxy velocities.Reinabelle Reyes, Rachel Mandelbaum, Uros Seljak, Tobias Baldauf, James E. Gunn, Lucas Lombriser, Robert E. Smith, . Nature 464:256-258,2010.

We also hear that Planck-scale Lorentz violation is still

possible. • Planck-scale Lorentz violation constrained by Ultra-

High-Energy Cosmic Rays.Luca Maccione, Andrew M. Taylor, David M. Mattingly, Stefano Liberati, . JCAP 0904:022,2009.

• Constraints on Lorentz invariance violation from gamma-ray burst GRB090510.Zhi Xiao, Bo-Qiang Ma, (Peking U.) .

Phys.Rev.D80:116005,2009.

There is some tendency of “dy-namical” dark energy.

There are various dark energy models.

Holographic Dark Energy ModelsAgegraphic Models

Entanglement Models

Exotic Matter Models

Modified Gravity Models: f(R) Gravity, …

What can we say about dark energy from Horava gravity ?

Planck (2012- )

PlanI. What is Horava gravity ?

II. Motivation of IR modification of Ho-rava gravity

III. FRW cosmology and dynamical dark energy in IR-modified Horava gravity

IV. Comparison with observational data

V. Open problems

VI. Final Remarks

I. What is Horava Gravity (‘09) ?

• It has been believed that Lorentz symmetry is a basic principle of our universe.

• But there have been continuous studies of Lorentz violation also, in extremely high energy.

• And also it seems that there is no reason of Lorentz symmetry at Planck energy, where the space-time would not be what we know. So, why not ?

• Once we abandon the Lorentz sym-metry at extremely high energy, like Planck energy, we can have (may be the first) testable quantum gravity, i.e., renormalizable, but without ghost.

• By abandoning the equal-footing treatment of space and time, Horava got a power-counting renormalizable gravity without ghost problem (‘09):

• “Splitting Time from Space”• “Standard Gravity Model ?”

II. Motivation of IR Modification of Horava Gravity

Renormalizable gravity theory by abandoning Lorentz symmetry in UV : Foliation Preserving Diffeomorphism.

Horava gravity ~ Einstein gravity (with a Lorentz deformation parameter )

+ non-covariant deformations with higher spa-tial derivatives (up to 6 orders)

+ “detailed balance” in the coefficients ( 5 constant parameters: )

Cf. Einstein gravity:

• The renormalizable quantum grav-ity can not be realized in Einstein’s gravity or its (relativistic) higher-derivature generalizations: There are ghosts, in addition to massless gravitons, and unitarity violation: In R+R^2 gravity, the full propaga-tor becomes

Massless gravitons Ghosts (!)

Why 6 order spatial deriva-tives ?

• But, for anisotropic (scaling) dimen-sions,

the propagator becomes(?)

At high energy with (z>1), this ex-pands as,

• Whereas at low energy,

G: Dimensionless coupling

Improved UV divergences but no ghost, i.e., no unitary problem.

Flow to z=1

Dimension counting• For an arbitrary spatial dimension D,

Dimensionless coupling for z=D: Power counting renormalizable

-D-z D+z

• z=3: Power counting renormaliz-able.

• z>3: Super-renromalizable.• Cf: (Newton’s) non-relativistic

gravity: z=2.• So, we need (unusual ?) RG-flows

as

k

z=3

z=1z=2

Detailed Balance Condi-tion:

• We need (foliation preserving Diff invariant) potential term having 6th order spatial deriva-tives at most (power-counting renormalizable with z=3) :

• There are large numbers of pos-sible terms, which are invariant by themselves, like …

• …, like

• But there are too many couplings for explicit computations, though some of them may be con-strained by the stability and uni-tarity. We need some pragmatic way of reducing in a reliable manner.

• Horava required the potential to be

by demanding

for some D-dimensional Euclidean ac-tion and the inverse of De Witt metric

• There is a similar method in non-equi-

librium critical phenomena.

• For D=3, W is 3-dimensional Eu-clidean action.

• First, we may consider Einstein-Hilbert action,

then, this gives 4’th-derivative order potential

• So, this is not enough to get 6’th order !!

• In 3-dim, we also have a peculiar, 3’rd- derivative order action, called (gravitational) Chern-Si-mons action.

• This produces the potential

with the Cotton tensor

Christoffel connection

• Then, in total, he got the 6’th or-der

from

So, we have 5 constant parame-ters, which seems to be minimum, from the detailed balancing.

• To summarize, the potential is given by the Detailed Balance be-tween terms:

• Analogy: a*x^2+b*x+c=a*(x+d)^2

• Some improved UV behaviors, without ghosts, are expected, i.e., renormalizabil-ity

Predictable Quantum Gravity !!(?)

• But, it seems that the detailed balance condition is too strong to get general spacetimes with an arbitrary cosmologi-cal constant.

• For example, there is no Minkowski , i.e., vanishing c.c. vacuum solution ! (Lu, Mei, Pope): There is no Newtonian grav-ity limit !!

• A “soft” breaking of the detailed bal-ance is given by the action :

• It is found that there does exit the black hole which converges to the usual Schwarzschild solution in Minkowski limit, i.e., for (s.t. Einstein-Hilbert in IR) (Kehagias, Sfetsos) .

IR modification term

• Black hole solution for limit ( ):

~ Schwarzshild Solution

: Independently of !!

General Remarks KS considered but it can

be considered as an independent pa-rameter: One more parameter than the Horava gravity with the detailed bal-ance, i.e., we have 6 constant parame-ters

• Cosmological constant ~ <0, i.e., AdS, for consistency ( >0 ) ! (Horava)

IR modification parameter

• dS , i.e., positive c.c., can be ob-tained by the continuation (Lu,Mei,Pope):

• Cf: KS:

III. FRW Cosmology and Dynamical Dark Energy in IR Modified Horava Gravity

• Homogeneous, isotropic cosmologi-cal solution of FRW form :

• For a perfect fluid with energy den-sity and pressure , the IR modi-fied Horava action gives …

Friedman equations

[ Upper (Lower) sign for AdS (dS) ]

is the current (a=1) radius of curvature of uni-verse

Remarks

• The term, which is the contribu-tion from the higher-derivative terms in Horava gravity, exists only for, , i.e., non-flat universe and becomes dominant for small : The cosmological solutions for GR are re-covered at large scales. (cf. Reyes, et al.)

• There is no contribution from the soft IR modification to the second Friedman Eq.: Identical to that of Lu,Mei,Pope.

What is the implication of the Horava gravity to our universe ?

What will we see if we have been lived in Horava gravity, from the begin-ning ?

• If we have been lived in the Ho-rava gravity (with some IR modi-fications), the additional contri-butions to the Friedman Eq. from the higher-(spatial) derivative terms may not be distinguish-able from the dark energy with (including C.C. term)

• We would see the Friedman Eq. as

where

• The Eq. of state parameter is given by

• And it depends on the constant parameters ...

IV a. Comparison with Observa-tional Data I

(1)Deceleration to Acceleration transition

Y. Gong, astro-ph/0405446:

• Actually, in our Horava gravity (the second Friedman Eq.), there is the transition point from de-celeration to acceleration phase, neglecting matter contributions, at

• If I use or ( ), I get

for the non-flat universe with .

Remarks

• At the transition point, the the-ory predicts , indepen-dently of the parameters !

(2) Non-flatness :

• If I use in the cur-rent epoch and

for the Hubble parameter ,

and , I get

If I use with , I get

To summarize,

• For , I get the constant pa-rameters with

which predicts the evolution of as one of the curves of .

• If I use from

and , I get

So, our theory predicts

• Or, in the astronomer’s conven-tion

PastDecelerationAcceleration

Y. Gong, astro-ph/0401207

Y. Gong, astro-ph/0405446

IV-b. Comparison with Observational Data : Latest Data, Without Knowing

Details of Matters.

• Previously, I neglected matters, which occupy about 30 % of our current universe, to get , so this would be good within about 70 % accuracy, only !

• Is there any more improved analy-sis to achieve better accuracy, without neglecting matters ? Yes ! …

• To this end, let me consider the series expansion of near the current epoch (a=1):

• This agrees exactly with Cheval-lier, Polarski, and Linder (CPL)'s parametrization !

• By knowing and from observational data, one can de-termine

as

Remarks

• I do not need to know about matter contents, separately.

• Once are determined, the whole function is com-pletely determined !

•

Data analysis without assuming the flat universe

Data analysis Ia, Ib: CMB+BAO+SN

• K. Ichikawa, T. Takahashi [arXiv: 0710.3995v2 [astro-ph] 3 May 2008 Ia

Ib

+Gold06 (red,solid): Analysis Ia

+David07 (blue,dotted) : Analysis Ib

Best Fit: (-1.10,0.39)

Best Fit: (-1.06,0.72)

Data analysis II: CMB+BAO+SN

• J.-Q.Xia, et. al., arXiv:0807.3878v2 [astro-ph] 22 Aug 2008

Non-Flat(blue, dash-dotted)

Flat (red, solid)

Best Fit: (-1.11,0.475)

The whole function of is deter-mined as (a=1/(1+z))

Future

Today

Past

Similar tendencies 1.

Best Fit: Gold-HST=142 SNe

U. Alam et. al., astro-ph/0403687 (Flat universe is assumed)

Similar tendencies 2Huterer and Cooray, PRD71, 023506 (2005): Uncorrealted estimates (flat universe is assumed)

SnIa

Similar tendencies (?) 2’Gong-Cai-Chen-Zhu, arXiv:0909.0596 :Uncorrealted estimates (flat universe is assumed)

SnIa+BAOIII+WMAP5+H(z)

Similar tendencies (?) 2’’Gong-Zhu-Zhu, arXiv:1008.5010  : Uncorrealted estimates (flat universe is assumed)

SnIa+BAO2+BAOz+WMAP7+H(z)

Similar tendencies (?) 2’’’

R. Amanullah et al. astro-ph/ 1004.1711  (flat universe is as-sumed)

Similar tendency 3

A. Shafieloo, astro-ph/0703034v3:SN Gold data set ( )

(flat universe is assumed)

Smooting method: Model independent !!(?)

Remark

• For the consistency of our theory, we need

• Otherwise, we would have imagi-nary valued and , though would not !! :

Consistency Conditions :

Forbidden !!Forbidden !!

In our data sets

Ia

IbII

Cosmologi-cal Con-stant

Within confidence levelsIa68.3 % Confidence

II

• Consistency condition may be tested near future, like in Planck (2012), by sharpening the data sets !

V. Open Problems• We need some more systematic fitting for

the range of allowed constant parameters to see whether our theory is really consistent with our universe.

• “Can we reproduce other complicated sto-ries with (dark) matters, i.e. density per-turbations ? “ (cf. A. Wang, et. al)

• Scale invariant Power spectrum with z=3

scalar field without inflation (Mukohyama et.al.): Inflation without inflation ??

VI. Final Remarks• 1. Dynamical dark energy is a

signal of Lorentz violation even in curvature dominated epoch:

• This means

Lorentz violating Unseen part in IR: Dynamical dark energy

• 2. Is the approach/result generic ?

• We have the same EOS for most gen-eral 4th-order spatial derivative terms (z=2).

• With the most general 6th-order terms (z=3), there are 1/a^6 term (stiff mat-ter) and this gives w=1 at a=0 (UV). But we do not need this to mimic our Universe.

• With the Detailed Balance, there is an-isotropic (local) Weyl invariance in UV and there is no 1/a^6 but 1/a^4 domi-nance (dark radiation) in UV.

• 3. Some possible non-triviality in matching with other early Uni-verse constraints (BBN, early ra-diation and matters).

• Early radiation, matter, and BBN data came from relativity or Newtonian particles. But it could be also modified due to UV Lorentz breaking (beyond curva-ture domi. Era).

• We expect some discrepancy with naïve application of known estimations based on relativity/Newtonian mechanics.

• 4. Implication to Horava gravity.

• Dynamical Dark energy is quite solid prediction of Horava grav-ity (unless we introduce some strange additional matters).

• So, this may be considered as a test of Horava gravity itself.