From satisfying to violating the null energy condition

Benjamin Elder, Austin Joyce, and Justin Khoury

Phys. Rev. D 89, 044027 – Published 19 February 2014

Abstract

We construct a theory that admits a time-dependent solution smoothly interpolating between a null energy condition (NEC)-satisfying phase at early times and a NEC-violating phase at late times. We first review earlier attempts to violate the NEC and an argument of Rubakov, presented in [V. A. Rubakov, Phys. Rev. D 88, 044015 (2013)], which forbids the existence of such interpolating solutions in a single-field dilation-invariant theory. We then construct a theory which, in addition to possessing a Poincaré-invariant vacuum, does admit such a solution. For a wide range of parameters, perturbations around this solution are at all times stable, comfortably subluminal and weakly coupled. The theory requires us to explicitly break dilation invariance, so it is unlikely that the theory is fully stable under quantum corrections, but we argue that the existence of a healthy interpolating solution is quantum-mechanically robust.

Received 3 December 2013

DOI:https://doi.org/10.1103/PhysRevD.89.044027

Authors & Affiliations

Benjamin Elder¹, Austin Joyce^1,2, and Justin Khoury¹

¹Department of Physics and Astronomy, Center for Particle Cosmology, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
²Enrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA

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Issue

Vol. 89, Iss. 4 — 15 February 2014

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Images

Figure 1
In a dilation invariant theory, we must have $Z = 0$ at both $e^{- 2 π} {\dot{π}}^{2} = 0$ and $e^{- 2 π} {\dot{π}}^{2} = Y$ , as well as $Z^{'} > 0$ at both of these points. It is impossible to connect these two solutions without having a region where $Z^{'} < 0$ , as is clear from the plot.
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Figure 2
NEC-satisfying and NEC-violating regions in the $(ϕ, ϕ^{'})$ phase space, for the parameter values $\frac{f_{\infty}}{f_{0}} = 10$ and $J_{0} = 0.75$ . The solution of interest, $ϕ = 1 + t / t_{*}$ , corresponding to $ϕ^{'} = 1$ , is plotted as a black dashed line. It first obeys the NEC for a period of time, and then crosses into the NEC-violating regime.
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Figure 3
Time line for the evolution. Our approximation of neglecting gravity is valid for the range $t_{beg} \leq t \leq t_{end}$ . For $t < t_{beg}$ , the universe asymptotes to a big bang singularity (since the NEC is satisfied in this regime). At approximately $t_{*}$ , the universe transitions from a NEC-satisfying phase to a NEC-violating one. For $t > t_{end}$ , cosmological expansion is important, and the universe must transition from the NEC-violating phase to a standard, radiation-dominated phase.
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Figure 4
The Hubble parameter is plotted for two values of $H_{i}$ with the fiducial parameters $J_{0} = 0.75$ , $f_{\infty} / f_{0} = 10$ , $Λ = 0.1$ , $t_{*} = - 1$ , and $C = - 6.3 \times 10^{- 4} M_{Pl}$ . The vertical dashed line marks the boundary between the NEC-satisfying and NEC-violating phases. As we would expect, $\dot{H} < 0$ when the NEC holds, and $\dot{H} > 0$ when it is violated. The dashed solution (corresponding to $H_{i} = 0.001 M_{Pl}$ ) always has $H > 0$ . It represents an initially expanding universe with decelerating expansion, and could match onto a big-bang-type solution for $t < t_{beg}$ . The solid line (with $H_{i} = - 0.001 M_{Pl}$ ) represents an initially contracting universe $(H < 0)$ which enters the phase of NEC-violation and undergoes a cosmological bounce to an accelerating phase $(H > 0)$ , all within the regime of validity of our effective theory.
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Figure 5
The shaded regions represent parts of the $(ϕ, ϕ^{'})$ phase space where perturbations (a) suffer from gradient instabilities, (b) propagate superluminally, and (c) are strongly coupled. The parameter values are $\frac{f_{\infty}}{f_{0}} = 10$ and $J_{0} = 0.75$ . The solution of interest, $ϕ = 1 + t / t_{*}$ , corresponding to $ϕ^{'} = 1$ , is plotted as a black dashed line. It avoids all pathological regions.
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Figure 6
Phase portrait with all constraints overlaid, again for the fiducial choice of parameters $\frac{f_{\infty}}{f_{0}} = 10$ and $J_{0} = 0.75$ . The shaded region represents the union of all pathological regions shown in Fig. 5. The green long-dashed line separates the NEC-satisfying and NEC-violating regions. The black short-dashed line corresponds to the background solution of interest, given by (49). The solid lines represent other background solutions (with different initial conditions).
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