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It has been pointed out that the null energy condition can be violated stably in some non-canonical scalar-field theories.
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P. Creminelli, M. A. Luty, A. Nicolis and L. Senatore, “Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies,” JHEP 0612
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2013
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For a review of inflation, see, e.g., J. Yokoyama, “Inflation: 1980-201X,” PTEP 2014
2014
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V. A. Rubakov, “The Null Energy Condition and its violation,” Phys. Usp. 57
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2012
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Y. Wang and R. Brandenberger, “Scale-Invariant Fluctuations from Galilean Genesis,” JCAP 1210
2012
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T. Kunimitsu and J. Yokoyama, “Higgs condensation as an unwanted curvaton,” Phys. Rev. D 86
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2013
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2014
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X. Gao, “Unifying framework for scalar-tensor theories of gravity,” Phys. Rev. D 90
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X. Gao, “Hamiltonian analysis of spatially covariant gravity,” Phys. Rev. D 90
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S. Nishi and T. Kobayashi, “Generalized Galilean Genesis,” JCAP 1503
2015
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