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Living with the Dark En-ergy in Horava Gravity
Mu-In ParkChonbuk Nat’al Univ.
Based on arXiv:0905.4480 [JHEP], arXiv:0906.4275 [J-CAP],
IEU-APCTP Workshop on Cos-mology and Fundamental Physics (18 May 2010, IEU)
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0. Outline1. Horava gravity and its IR modi-
fication
2. FRW cosmology in IR modified Horava gravity
3. Comparison with observational data
4. Open problems
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1. 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:
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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 …
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• …, 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.
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• Horava required the potential to be of
by demanding
for some D-dimensional action and the inverse of De Witt metric
• There is a similar method in non-equi-
librium critical phenomena.
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• W is 3-dimensional Euclidean ac-tion.
• 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 !!
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• 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
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• 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.
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• 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 !!
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• 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
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• Black hole solution for limit ( ):
~ Schwarzshild Solution
: Independently of !!
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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
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• dS , i.e., positive c.c., can be ob-tained by the continuation (Lu,Mei,Pope):
• Cf: KS:
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2. FRW cosmology in IR modified Horava grav-ity
• 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 …
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Friedman equations
[ Upper (Lower) sign for AdS (dS) ]
is the current (a=1) radius of curvature of uni-verse
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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 cosmologi-cal solutions for GR are recovered at large scales. (cf. Reyes, et al.)
• There is no contribution from the soft IR modification to the second Fried-man Eq.: Identical to that of Lu,Mei,Pope.
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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 ?
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• 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)
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• We would see the Friedman Eq. as
where
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• The Eq. of state parameter is given by
• And it depends on the constant parameters ...
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3a. 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 accu-racy, without neglecting matters ? Yes ! …
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• 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 !
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• By knowing and from observational data, one can de-termine
as
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Remarks
• I do not need to know about matter contents, separately.
• Once are determined, the whole function is com-pletely determined !
•
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Data analysis without assuming the flat universe
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Data analysis Ia, Ib: CMB+BAO+SN
• K. Ichikawa, T. Takahashi [arXiv: 0710.3995v2 [astro-ph] 3 May 2008 Ia
Ib
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+Gold06 (red,solid): Analysis Ia
+David07 (blue,dotted) : Analysis Ib
Best Fit: (-1.10,0.39)
Best Fit: (-1.06,0.72)
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Data analysis II: CMB+BAO+SN
• J.-Q.Xia, et. al., arXiv:0807.3878v2 [astro-ph] 22 Aug 2008
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Non-Flat(blue, dash-dotted)
Flat (red, solid)
Best Fit: (-1.11,0.475)
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The whole function of is deter-mined as (a=1/(1+z))
Future
Today
Past
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Similar tendencies 1.
Best Fit: Gold-HST=142 SNe
U. Alam et. al., astro-ph/0403687 (Flat universe is asumed)
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Similar tendencies 2Huterer and Cooray, PRD71, 023506 (2005): Uncorrealted estimates (flat universe is assumed)
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Similar tendencies (?) 2’
R. Amanullah et al. astro-ph/ 1004.1711 (flat universe is as-sumed)
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Similar tendency 3
Shafieloo, astro-ph/0703034v3:SN Gold data set ( )
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Remark
• For the consistency of our theory, we need
• Otherwise, we would have imagi-nary valued and , though would not !! :
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Consistency Conditions :
Forbidden !!Forbidden !!
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In our data sets
Ia
IbII
Cosmologi-cal Con-stant
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Within confidence levelsIa68.3 % Confidence
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II
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• Consistency condition may be tested near future, like in Planck (2012), by sharpening the data sets !
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4. Open problems• We need some more systematic fitting
for the range of allowed constant pa-rameters to see whether our theory is really consistent with our uni-verse.
• “Can we reproduce other complicated stories with (dark) matters, i.e. density perturbations ? “ (cf. A. Wang, et. al)
• Inflation without inflation ??