Living with the Dark Energy in Horava Gravity Mu-In Park Kunsan Nat’al Univ., Korea Based on arXiv: [JHEP], arXiv: [JCAP], APCTP-IEU Focus Program on Cosmology and Fundamental Physics 2 (8 June 2011)

55) Quantum Gravity at a Lifshitz Point. Petr Horava, (UC, Berkeley & LBL, Berkeley). Jan (Published Jan 2009). 29pp. Published in Phys.Rev.D79:084008,2009. e-Print: arXiv: [hep-th] Petr HoravaUC, BerkeleyLBL, Berkeley TOPCITE = 250+ Cited 474 times 474 times 66) Membranes at Quantum Criticality. Petr Horava, (UC, Berkeley & LBL, Berkeley). Dec pp. Published in JHEP 0903:020,2009. e-Print: arXiv: [hep-th] Petr HoravaUC, BerkeleyLBL, Berkeley TOPCITE = 100+ Cited 258 times258 times 11) General Covariance in Quantum Gravity at a Lifshitz Point. Petr Horava, Charles M. Melby-Thompson. Jul (Published Sep 15, 2010). 41pp. Published in Phys.Rev.D82:064027,2010. e-Print: arXiv: [hep-th] Cited 50 times Petr HoravaCharles M. Melby-Thompson 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 experimentation -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: ,2010. Reinabelle ReyesRachel MandelbaumUros SeljakTobias BaldaufJames E. GunnLucas LombriserRobert E. Smith

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. Luca MaccioneAndrew M. TaylorDavid M. MattinglyStefano Liberati Constraints on Lorentz invariance violation from gamma-ray burst GRB Zhi Xiao, Bo-Qiang Ma, (Peking U.). Zhi XiaoBo-Qiang MaPeking U. Phys.Rev.D80:116005,2009.

There is some tendency of “dynamical” dark energy.

There are various dark energy models. Holographic Dark Energy Models Agegraphic Models Entanglement Models Exotic Matter Models Modified Gravity Models: f(R) Gravity, …

What can we say about dark energy from Horava gravity ? Planck (2012- )

Plan I. What is Horava gravity ? II. Motivation of IR modification of Horava 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 symmetry 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 spatial derivatives (up to 6 orders) + “detailed balance” in the coefficients ( 5 constant parameters: ) Cf. Einstein gravity:

The renormalizable quantum gravity 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 propagator becomes Massless gravitons Ghosts (!) Why 6 order spatial derivatives ?

But, for anisotropic (scaling) dimensions, the propagator becomes(?) At high energy with (z>1), this expands 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 renormalizable. z>3: Super-renromalizable. Cf: (Newton’s) non-relativistic gravity: z=2. So, we need (unusual ?) RG-flows as k z=3 z=1 z=2

Detailed Balance Condition: We need (foliation preserving Diff invariant) potential term having 6 th order spatial derivatives at most (power-counting renormalizable with z=3) : There are large numbers of possible terms, which are invariant by themselves, like …

…, like But there are too many couplings for explicit computations, though some of them may be constrained by the stability and unitarity. We need some pragmatic way of reducing in a reliable manner.

Horava required the potential to be by demanding for some D-dimensional Euclidean action and the inverse of De Witt metric There is a similar method in non-equilibrium critical phenomena.

For D=3, W is 3-dimensional Euclidean 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-Simons action. This produces the potential with the Cotton tensor Christoffel connection

Then, in total, he got the 6’th order from So, we have 5 constant parameters, which seems to be minimum, from the detailed balancing.

To summarize, the potential is given by the Detailed Balance between terms: Analogy: a*x^2+b*x+c=a*(x+d)^2

Some improved UV behaviors, without ghosts, are expected, i.e., renormalizability Predictable Quantum Gravity !!(?) But, it seems that the detailed balance condition is too strong to get general spacetimes with an arbitrary cosmological constant. For example, there is no Minkowski, i.e., vanishing c.c. vacuum solution ! (Lu, Mei, Pope): There is no Newtonian gravity limit !!

A “soft” breaking of the detailed balance 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 parameter: One more parameter than the Horava gravity with the detailed balance, i.e., we have 6 constant parameters Cosmological constant ~ 0 ) ! (Horava) IR modification parameter

dS, i.e., positive c.c., can be obtained by the continuation (Lu,Mei,Pope): Cf: KS:

III. FRW Cosmology and Dynamical Dark Energy in IR Modified Horava Gravity Homogeneous, isotropic cosmological solution of FRW form : For a perfect fluid with energy density and pressure, the IR modified Horava action gives …

Friedman equations [ Upper (Lower) sign for AdS (dS) ] is the current (a=1) radius of curvature of universe

Remarks The term, which is the contribution 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 recovered 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 beginning ?

If we have been lived in the Horava gravity (with some IR modifications), the additional contributions to the Friedman Eq. from the higher-(spatial) derivative terms may not be distinguishable 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 Observational Data I (1)Deceleration to Acceleration transition

Y. Gong, astro-ph/ :

Actually, in our Horava gravity (the second Friedman Eq.), there is the transition point from deceleration 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 theory predicts, independently of the parameters !

(2) Non-flatness :

If I use in the current epoch and for the Hubble parameter, and, I get If I use with, I get

To summarize, For, I get the constant parameters 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 convention Past Deceleration Acceleration

Y. Gong, astro-ph/

Y. Gong, astro-ph/

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 analysis 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 Chevallier, Polarski, and Linder (CPL)'s parametrization !

By knowing and from observational data, one can determine as

Remarks I do not need to know about matter contents, separately. Once are determined, the whole function is completely determined !

Data analysis without assuming the flat universe

Data analysis Ia, Ib: CMB+BAO+SN K. Ichikawa, T. Takahashi [arXiv: v2 [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: v2 [astro-ph] 22 Aug 2008

Non-Flat (blue, dash-dotted) Flat (red, solid) Best Fit: (-1.11,0.475)

The whole function of is determined as (a=1/(1+z)) Future Today Past

Similar tendencies 1. Best Fit: Gold-HST=142 SNe U. Alam et. al., astro-ph/ (Flat universe is assumed)

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

Similar tendencies (?) 2’ Gong-Cai-Chen-Zhu, arXiv: : Uncorrealted estimates (flat universe is assumed) SnIa+BAOIII+WMAP5+H(z)

Similar tendencies (?) 2’’ GongGong-Zhu-Zhu, arXiv: :Zhu Uncorrealted estimates (flat universe is assumed) SnIa+BAO2+BAOz+WMAP7+H(z)

Similar tendencies (?) 2’’’ R. Amanullah et al. astro-ph/ (flat universe is assumed)

Similar tendency 3 A. Shafieloo, astro-ph/ v3: 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 imaginary valued and, though would not !! :

Consistency Conditions : Forbidden !!

In our data sets Ia Ib II Cosmological Constant

Within confidence levels Ia 68.3 % 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 stories with (dark) matters, i.e. density perturbations ? “ (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 general 4th-order spatial derivative terms (z=2). With the most general 6 th -order terms (z=3), there are 1/a^6 term (stiff matter) 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 anisotropic (local) Weyl invariance in UV and there is no 1/a^6 but 1/a^4 dominance (dark radiation) in UV.

3. Some possible non-triviality in matching with other early Universe constraints (BBN, early radiation 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 curvature 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 gravity (unless we introduce some strange additional matters). So, this may be considered as a test of Horava gravity itself.