Abstract
In this paper, we study inflation in the general covariant Hořava-Lifshitz gravity without the projectability condition. We write down explicitly the equations of the linear scalar perturbations of the FRW universe for a single scalar field without specifying to any gauge. Applying these equations to a particular gauge, we are able to obtain a master equation of the perturbations, in contrast to all the other versions of the theory without the projectability condition. This is because in the current version of the theory it has the same degree of freedom of general relativity. To calculate the power spectrum and index, we first define the initial conditions as the ones that minimize the energy of the ground state. Then, we obtain the full solutions of the equation of motion. From these solutions, we calculate the power spectrum and spectrum index of the comoving curvature perturbations and find the corrections due to the high order spatial derivative terms of the theory to those standard results obtained in general relativity. Remarkably, partly of corrections are a direct consequence of the non-projectability condition. It is also shown that the perturbations are still of scale-invariance, and the results obtained in the general covariant Hořava-Lifshitz gravity without the projectability condition are consistent with all current cosmological observations.
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References
P. Hořava, Membranes at Quantum Criticality, JHEP 03 (2009) 020 [arXiv:0812.4287] [INSPIRE].
P. Hořava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
P. Hořava, Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point, Phys. Rev. Lett. 102 (2009) 161301 [arXiv:0902.3657] [INSPIRE].
M. Visser, Lorentz symmetry breaking as a quantum field theory regulator, Phys. Rev. D 80 (2009) 025011 [arXiv:0902.0590] [INSPIRE].
M. Visser, Power-counting renormalizability of generalized Hořava gravity, arXiv:0912.4757 [INSPIRE].
C. Germani, A. Kehagias and K. Sfetsos, Relativistic quantum gravity at a Lifshitz point, JHEP 09 (2009) 060 [arXiv:0906.1201] [INSPIRE].
D. Blas, O. Pujolàs and S. Sibiryakov, Consistent Extension of Hořava Gravity, Phys. Rev. Lett. 104 (2010) 181302 [arXiv:0909.3525] [INSPIRE].
D. Blas, O. Pujolàs and S. Sibiryakov, Models of non-relativistic quantum gravity: the good, the bad and the healthy, JHEP 04 (2011) 018 [arXiv:1007.3503] [INSPIRE].
I. Kimpton and A. Padilla, Lessons from the decoupling limit of Hořava gravity, JHEP 07 (2010) 014 [arXiv:1003.5666] [INSPIRE].
C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, W.H. Freeman and Company, San Francisco U.S.A. (1973), pg. 484.
H. Lü, J. Mei and C. Pope, Solutions to Hořava gravity, Phys. Rev. Lett. 103 (2009) 091301 [arXiv:0904.1595] [INSPIRE].
G. Calcagni, Cosmology of the Lifshitz universe, JHEP 09 (2009) 112 [arXiv:0904.0829] [INSPIRE].
R.-G. Cai, L.-M. Cao and N. Ohta, Topological Black Holes in Hořava-Lifshitz Gravity, Phys. Rev. D 80 (2009) 024003 [arXiv:0904.3670] [INSPIRE].
A. Kehagias and K. Sfetsos, The black hole and FRW geometries of non-relativistic gravity, Phys. Lett. B 678 (2009) 123 [arXiv:0905.0477] [INSPIRE].
M.-I. Park, The black hole and cosmological solutions in IR modified Hořava gravity, JHEP 09 (2009) 123 [arXiv:0905.4480] [INSPIRE].
A. Ghodsi and E. Hatefi, Extremal rotating solutions in Hořava gravity, Phys. Rev. D 81 (2010) 044016 [arXiv:0906.1237] [INSPIRE].
K. Izumi and S. Mukohyama, Stellar center is dynamical in Hořava-Lifshitz gravity, Phys. Rev. D 81 (2010) 044008 [arXiv:0911.1814] [INSPIRE].
E. Kiritsis, Spherically symmetric solutions in modified Hořava-Lifshitz gravity, Phys. Rev. D 81 (2010) 044009 [arXiv:0911.3164] [INSPIRE].
G. Koutsoumbas, E. Papantonopoulos, P. Pasipoularides and M. Tsoukalas, Black Hole Solutions in 5D Hořava-Lifshitz Gravity, Phys. Rev. D 81 (2010) 124014 [arXiv:1004.2289] [INSPIRE].
P. Hořava, General Covariance in Gravity at a Lifshitz Point, Class. Quant. Grav. 28 (2011) 114012 [arXiv:1101.1081] [INSPIRE].
A. Borzou, K. Lin and A. Wang, Detailed balance condition and ultraviolet stability of scalar field in Hořava-Lifshitz gravity, JCAP 05 (2011) 006 [arXiv:1103.4366] [INSPIRE].
T.P. Sotiriou, M. Visser and S. Weinfurtner, Phenomenologically viable Lorentz-violating quantum gravity, Phys. Rev. Lett. 102 (2009) 251601 [arXiv:0904.4464] [INSPIRE].
T.P. Sotiriou, M. Visser and S. Weinfurtner, Quantum gravity without Lorentz invariance, JHEP 10 (2009) 033 [arXiv:0905.2798] [INSPIRE].
E. Kiritsis and G. Kofinas, Hořava-Lifshitz cosmology, Nucl. Phys. B 821 (2009) 467 [arXiv:0904.1334] [INSPIRE].
A. Wang and R. Maartens, Linear perturbations of cosmological models in the Hořava-Lifshitz theory of gravity without detailed balance, Phys. Rev. D 81 (2010) 024009 [arXiv:0907.1748] [INSPIRE].
A. Padilla, The good, the bad and the ugly of Hořava gravity, J. Phys. Conf. Ser. 259 (2010) 012033 [arXiv:1009.4074] [INSPIRE].
T.P. Sotiriou, Hořava-Lifshitz gravity: a status report, J. Phys. Conf. Ser. 283 (2011) 012034 [arXiv:1010.3218] [INSPIRE].
T. Clifton, P.G. Ferreira, A. Padilla and C. Skordis, Modified gravity and cosmology, Phys. Rept. 513 (2012) 1 [arXiv:1106.2476] [INSPIRE].
S. Mukohyama, Hořava-Lifshitz cosmology: a review, Class. Quant. Grav. 27 (2010) 223101 [arXiv:1007.5199] [INSPIRE].
C. Bogdanos and E.N. Saridakis, Perturbative instabilities in Hořava gravity, Class. Quant. Grav. 27 (2010) 075005 [arXiv:0907.1636] [INSPIRE].
Y. Huang, A. Wang and Q. Wu, Stability of the de Sitter spacetime in Hořava-Lifshitz theory, Mod. Phys. Lett. A 25 (2010) 2267 [arXiv:1003.2003] [INSPIRE].
A. Wang and Q. Wu, Stability of spin-0 graviton and strong coupling in Hořava-Lifshitz theory of gravity, Phys. Rev. D 83 (2011) 044025 [arXiv:1009.0268] [INSPIRE].
C. Charmousis, G. Niz, A. Padilla and P.M. Saffin, Strong coupling in Hořava gravity, JHEP 08 (2009) 070 [arXiv:0905.2579] [INSPIRE].
D. Blas, O. Pujolàs and S. Sibiryakov, On the extra mode and inconsistency of Hořava gravity, JHEP 10 (2009) 029 [arXiv:0906.3046] [INSPIRE].
K. Koyama and F. Arroja, Pathological behaviour of the scalar graviton in Hořava-Lifshitz gravity, JHEP 03 (2010) 061 [arXiv:0910.1998] [INSPIRE].
A. Papazoglou and T.P. Sotiriou, Strong coupling in extended Hořava-Lifshitz gravity, Phys. Lett. B 685 (2010) 197 [arXiv:0911.1299] [INSPIRE].
A. Vainshtein, To the problem of nonvanishing gravitation mass, Phys. Lett. B 39 (1972) 393 [INSPIRE].
V.A. Rubakov and P.G. Tinyakov, Infrared-modified gravities and massive gravitons, Phys. Usp. 51 (2008) 759.
K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].
K. Izumi and S. Mukohyama, Nonlinear superhorizon perturbations in Hořava-Lifshitz gravity, Phys. Rev. D 84 (2011) 064025 [arXiv:1105.0246] [INSPIRE].
A.E. Gumrukcuoglu, S. Mukohyama and A. Wang, General relativity limit of Hořava-Lifshitz gravity with a scalar field in gradient expansion, Phys. Rev. D 85 (2012) 064042 [arXiv:1109.2609] [INSPIRE].
T. Zhu, Q. Wu, A. Wang and F.-W. Shu, U(1) symmetry and elimination of spin-0 gravitons in Hořava-Lifshitz gravity without the projectability condition, Phys. Rev. D 84 (2011) 101502 [arXiv:1108.1237] [INSPIRE].
T. Zhu, F.-W. Shu, Q. Wu and A. Wang, General covariant Hořava-Lifshitz gravity without projectability condition and its applications to cosmology, Phys. Rev. D 85 (2012) 044053 [arXiv:1110.5106] [INSPIRE].
P. Hořava and C.M. Melby-Thompson, General Covariance in Quantum Gravity at a Lifshitz Point, Phys. Rev. D 82 (2010) 064027 [arXiv:1007.2410] [INSPIRE].
A. Wang and Y. Wu, Cosmology in nonrelativistic general covariant theory of gravity, Phys. Rev. D 83 (2011) 044031 [arXiv:1009.2089] [INSPIRE].
Y. Huang and A. Wang, Stability, ghost and strong coupling in nonrelativistic general covariant theory of gravity with λ = 1, Phys. Rev. D 83 (2011) 104012 [arXiv:1011.0739] [INSPIRE].
A.M. da Silva, An Alternative Approach for General Covariant Hořava-Lifshitz Gravity and Matter Coupling, Class. Quant. Grav. 28 (2011) 055011 [arXiv:1009.4885] [INSPIRE].
J. Kluson, Hamiltonian Analysis of Non-Relativistic Covariant RFDiff Hořava-Lifshitz Gravity, Phys. Rev. D 83 (2011) 044049 [arXiv:1011.1857] [INSPIRE].
K. Lin, A. Wang, Q. Wu and T. Zhu, On strong coupling in nonrelativistic general covariant theory of gravity, Phys. Rev. D 84 (2011) 044051 [arXiv:1106.1486] [INSPIRE].
Y. Huang, A. Wang and Q. Wu, Inflation in general covariant theory of gravity, JCAP 10 (2012) 010 [arXiv:1201.4630] [INSPIRE].
J. Greenwald, V. Satheeshkumar and A. Wang, Black holes, compact objects and solar system tests in non-relativistic general covariant theory of gravity, JCAP 12 (2010) 007 [arXiv:1010.3794] [INSPIRE].
J. Greenwald, J. Lenells, J. Lu, V. Satheeshkumar and A. Wang, Black holes and global structures of spherical spacetimes in Hořava-Lifshitz theory, Phys. Rev. D 84 (2011) 084040 [arXiv:1105.4259] [INSPIRE].
A. Borzou, K. Lin and A. Wang, Static electromagnetic fields and charged black holes in general covariant theory of Hořava-Lifshitz gravity, JCAP 02 (2012) 025 [arXiv:1110.1636] [INSPIRE].
J. Alexandre and P. Pasipoularides, Spherically symmetric solutions in Covariant Hořava-Lifshitz Gravity, Phys. Rev. D 83 (2011) 084030 [arXiv:1010.3634] [INSPIRE].
J. Alexandre and P. Pasipoularides, Spherically symmetric solutions, Newton’s Law and the infrared limit λ → 1 in covariant Hořava-Lifshitz gravity, Phys. Rev. D 84 (2011) 084020 [arXiv:1108.1348].
K. Lin, S. Mukohyama and A. Wang, Solar system tests and interpretation of gauge field and Newtonian prepotential in general covariant Hořava-Lifshitz gravity, Phys. Rev. D 86 (2012) 104024 [arXiv:1206.1338] [INSPIRE].
X. Gao, Y. Wang, R. Brandenberger and A. Riotto, Cosmological Perturbations in Hořava-Lifshitz Gravity, Phys. Rev. D 81 (2010) 083508 [arXiv:0905.3821] [INSPIRE].
B. Chen, S. Pi and J.-Z. Tang, Scale Invariant Power Spectrum in Hořava-Lifshitz Cosmology without Matter, JCAP 08 (2009) 007 [arXiv:0905.2300] [INSPIRE].
T. Kobayashi, Y. Urakawa and M. Yamaguchi, Cosmological perturbations in a healthy extension of Hořava gravity, JCAP 04 (2010) 025 [arXiv:1002.3101] [INSPIRE].
A. Cerioni and R.H. Brandenberger, Cosmological Perturbations in the ’Healthy Extension’ of Hořava-Lifshitz gravity, arXiv:1008.3589 [INSPIRE].
R.-G. Cai, B. Hu and H.-B. Zhang, Scalar graviton in the healthy extension of Hořava-Lifshitz theory, Phys. Rev. D 83 (2011) 084009 [arXiv:1008.5048] [INSPIRE].
E.G. Ferreira and R. Brandenberger, The Trans-Planckian Problem in the Healthy Extension of Hořava-Lifshitz Gravity, Phys. Rev. D 86 (2012) 043514 [arXiv:1204.5239] [INSPIRE].
A. Wang, D. Wands and R. Maartens, Scalar field perturbations in Hořava-Lifshitz cosmology, JCAP 03 (2010) 013 [arXiv:0909.5167] [INSPIRE].
K.A. Malik and D. Wands, Cosmological perturbations, Phys. Rept. 475 (2009) 1 [arXiv:0809.4944] [INSPIRE].
D. Baumann, TASI Lectures on Inflation, arXiv:0907.5424 [INSPIRE].
J. Martin and R.H. Brandenberger, The TransPlanckian problem of inflationary cosmology, Phys. Rev. D 63 (2001) 123501 [hep-th/0005209] [INSPIRE].
J. Martin and R. Brandenberger, On the dependence of the spectra of fluctuations in inflationary cosmology on transPlanckian physics, Phys. Rev. D 68 (2003) 063513 [hep-th/0305161] [INSPIRE].
J. Martin and R. Brandenberger, On the dependence of the spectra of fluctuations in inflationary cosmology on transPlanckian physics, Phys. Rev. D 68 (2003) 063513 [hep-th/0305161] [INSPIRE].
J.C. Niemeyer and R. Parentani, Transplanckian dispersion and scale invariance of inflationary perturbations, Phys. Rev. D 64 (2001) 101301 [astro-ph/0101451] [INSPIRE].
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Zhu, T., Huang, Y. & Wang, A. Inflation in general covariant Hořava-Lifshitz gravity without projectability. J. High Energ. Phys. 2013, 138 (2013). https://doi.org/10.1007/JHEP01(2013)138
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DOI: https://doi.org/10.1007/JHEP01(2013)138