Effective Field Theory of inflationin Horava Lifshitz gravity
Shun Arai (Nagoya University) arai.shun@a.mbox.nagoya-u.ac.jp
collaborators: Sergey Sibiryakov (CERN, INR RAS), Yuko Urakawa(Nagoya University)
Abstract
We study the evolution of scalar perturbations in inflationary epoch with a single Lifshitz scalar in the context of the BPSH theory,
which generalizes the original non-projectable Horava Lifshitz gravity. In the previous studies, power spectrum of the curvaturein the fixed de-Sitter background, the evolution in the IR regime and the evolution the isocurvature mode in the Einstein Aether
theory have been obtained. In our study, we consistently solve the evolution of the coupled scalar graviton and the inflaton
fluctuation and we obtain only the adiabatic perturbation is generated because the scalar graviton become massive Hubble
crossing. Moreover, we show that the detection of primordial gravitational waves can verify LV in inflationary epoch.

1.Introduction
Purposes of our study

Horava Lifshitz gravity (HL gravity) is the one of the modified theory of gravity
which is power-counting renormalizable in the UV regime(P.Horava 2009).
2Mpl
SEH =2GRZ

2
Mpl
SHL =
2HL gravity

Z4d x

dtd3 x2Mpl
gR !
2pp

N {(Kij K ij

anisotropic scalingzt ! b t, x ! bx=) h ! bZ

d4 x{(@h)2 + h(@h)2 + · · · }dimh = 1・Investigate consistent growth of the scalar graviton and theinflaton fluctuation in the Lifshitz regime.

irrelevant term

・Derive the curvature perturbation and isocurvature

K 2 ) + R(3) + V}

perturbation.

R, R2 , Rij Rij , (ri R)2 , (ri Rjk )2(3 z)/2hdimh =3z2

 0 (z3)・Compare the curvature & isocurvature power spectra with

But the anisotropic scaling law violates the local Lorentz symmetry in the UV
regime. Then a new scalar degree of freedom appears called “scalar graviton”
or “khronon”.

current CMB observation to constrain to LV scale andmodulations of higher derivatives.

2.Formulation
Then we take the metric ansatz of perturbation

We introduce the general action of the BPSH theory including thetransition energy scale M* where the local Lorentz symmetry is broken.
Z
1 2
1p M⇤2 1
SHL = dtd3 xN
Kij K ij
K +R + ai ai2↵↵↵
123Gravity


1 Rij RijR2 Rri ai ai ai++2ds = N dt +(dx + N dt)(dx + N dt)
1
2
341

K =( ˙5N5 N)
2N
221(ri Rjk )(ri R)
Rri ai ai 2 ai
a = @ lnN non-projectable condition+R
3-dimensional Ricci tensor
2M⇤212
343-dimensional Laplacian"Z
23p ( ˙ N i @i )2Inflaton S = 1
dtd3 xN g+ 1
2+ 32222
2N22M⇤
2M⇤ 4
sector

Metric ansatz
N = e , Ni = @i B, ij = a2 e2R ij

sector

22ij

i

2ij

i

iji

jijj

R : scalar graviton ' : inflaton fluctuation

jiiij#V( ) .

momentum & energy
constraints

Inflaton is a single Lifshitz scalar field and thus higher derivative terms
appear in the UV regime.

Then we compute the dynamics of 2 scalars thatinteract each other.

3.Results
3.1 initial density fluctuations3.2 Smoking gun for Lorentz Violation (LV)

In some energy scale higher than M*, the scalar
graviton become massive due to non-locality ofHamltonian constraint with its mass as

Inflation models with 4D Diffs invariance, the consistency relation between scalarand tensor fluctuations are generally given by

1mK = p H
↵HSince the mass is much larger than Hubble scale,an adiabatic perturbation can be generated asHH'⇠'˙˙with its power spectrum as⇣⌘R↵1P⇣ /"✓HpM⇤nt =◆3/z 1r8cswhich is always negative nonzero value. On the contrary in Horava Lifshitz
gravity, the consistency relation is modified due to LV as

nt /3z rz 16

nt = 0 (z = 3)

which means we can obtain a direct evidence of LV if we detect primordial
gravitational waves with totally scale invariant spectrum.

4.Summary & Future works

We have obtained the solutions of the scalar graviton and inflaton fluctuation. But our analysis has yet to complete among the regime
where the curvature fluctuation conserves. Therefore we will have to calculate the perturbations in the whole regimes and obtain thecurvature power spectrum. Then we will obtain constraints of the LV scale and modulations of higher derivatives by using thePLANCK data. Moreover, the detection of primordial gravitational waves can verify LV in inflationary epoch.

5.References

P. Horava, Phys. Rev. D 79, 084008 (2009) [arXiv:0901.3775 [hep-th]]. P. Creminelli, J. Gleyzes, J. Norena and F. Vernizzi, Phys. Rev. Lett. 113, no. 23, 231301 (2014)[arXiv:1407.8439 [astro-ph.CO]].
D. Blas, O. Pujolas and S. Sibiryakov, JHEP 1104, 018 (2011) [arXiv:1007.3503 [hep-th]]. C.Armendariz-Picon, N.F.Sierra, J.Garriga. JCAP 1007.010,2010[arXiv:1003.1283 [hep-th]]