From Satisfying to Violating the Null Energy Condition
FFrom Satisfying to Violating the Null EnergyCondition
Benjamin Elder a , Austin Joyce a , b and Justin Khoury a a Center for Particle Cosmology, Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, PA 19104 b Enrico Fermi Institute and Kavli Institute for Cosmological Physics,University of Chicago, Chicago, IL 60637
Abstract
We construct a theory which admits a time-dependent solution smoothly interpolating between a null energycondition (NEC)-satisfying phase at early times and a NEC-violating phase at late times. We first reviewearlier attempts to violate the NEC and an argument of Rubakov, presented in 1305.2614, which forbidsthe existence of such interpolating solutions in a single-field dilation-invariant theory. We then constructa theory which, in addition to possessing a Poincar´e-invariant vacuum, does admit such a solution. For awide range of parameters, perturbations around this solution are at all times stable, comfortably subluminaland weakly-coupled. The theory requires us to explicitly break dilation-invariance, so it is unlikely that thetheory is fully stable under quantum corrections, but we argue that the existence of a healthy interpolatingsolution is quantum-mechanically robust.
Energy conditions are usually imposed for convenience, based upon our expectations for how mattershould behave. In particular, they are covariantizations of the notion that energy density should bea positive quantity. Of all the energy conditions, the
Null Energy Condition (NEC), which statesthat T µν n µ n ν ≥ , (1)for any null vector n µ , appears to be the most fundamental. Unlike other energy conditions, itcannot be violated by the addition of a suitably large vacuum energy contribution, so in this senseit is an unambiguous constraint on the matter. Moreover, in Einstein gravity, the NEC is necessaryto establish the second law of black hole thermodynamics [1]. Thirdly, in cosmology the NECprecludes a non-singular bounce. Assuming spatial flatness, the Hubble parameter satisfies M ˙ H = −
12 ( ρ + P ) . (2)For a perfect fluid, the NEC implies ρ + P ≥
0, and thus ˙ H ≤
0. Contraction (
H <
0) cannot evolveto expansion (
H > a r X i v : . [ h e p - t h ] J a n athologies [63], including ghosts, gradient instabilities, superluminality, absence of a Lorentz-invariant vacuum, etc . Progress has been made in avoiding some of these shortcomings [53, 54, 59–61, 65], as reviewed below (see Table 1), though a fully satisfactory example remains elusive. Itis important to push this program further, to sharpen the connection between the NEC and thestandard assumptions of quantum field theory.The
DBI Genesis scenario [61], based on the DBI conformal galileons [66], is the closest any theoryhas come to achieving NEC violation while satisfying the standard properties of a local quantumfield theory. Specifically, as shown in [61], the coefficients of the five DBI galileon terms can bechosen such that:1. The theory admits a stable, Poincar´e-invariant vacuum. Further, the Lorentz-invariant S-matrix about this vacuum obeys the simplest dispersion relations for 2 → exact solution ofthe effective theory, including all possible higher-dimensional operators consistent with theassumed symmetries.3. Perturbations around the NEC-violating background, and around small deformations thereof,propagate subluminally.4. This solution is stable against radiative corrections and the effective theory for perturbationsabout this solution is well-defined.This represents a significant improvement over ghost condensation [67] (which fails to satisfy 1) andthe ordinary conformal galileons [54, 68] (which fail to satisfy 1 and 3). Additionally, consistencywith black hole thermodynamics is desirable [69]. This remains an open issue which deserves furtherstudy. It is worth pointing out that the non-minimal couplings to gravity inherent in the theorywill modify the usual link between NEC violation and the black hole area law.Unfortunately, the DBI Genesis theory itself suffers from two drawbacks. Similar to the conformalgalileons, one can find weak deformations of the Poincar´e-invariant solution around which pertur-bations propagate superluminally. As pointed out recently [70, 71], however, galileon theories thatadmit superluminality can sometimes be mapped through field redefinitions to healthy galileon the-ories, indicating that the apparent superluminalty is unphysical. So in this sense superluminalitydoes not offer a clear-cut criterion. But it would certainly be preferable to have an example wheresuperluminality is manifestly absent.A less ambiguous drawback was pointed out by Rubakov [65]: although the theory admits bothPoincar´e-invariant and NEC-violating solutions, any solution that attempts to interpolate betweenthe two vacua inevitably hits a strong coupling point. In other words, the kinetic term of fluctuationsaround any interpolating solution goes to zero somewhere. In particular, it is impossible to create For example, the Hamiltonian of theories which violate the NEC was argued to be unbounded from below in [64]. Note that the conformal galileon Lagrangian can be deformed in a straightforward way to remove superluminalpropagation about the NEC-violating background [59]. We thank Claudia de Rham and Andrew Tolley for a discussion on this point. The argument, which we review in Sec. 3, is quitegeneral — it only assumes that the theory describes a single field which is dilation-invariant, andthat both Poincar´e and NEC-violating solutions preserve this symmetry. Rubakov showed that theconclusions can be evaded by introducing additional scalar fields.In this paper, we stick to a single-field theory but relax the assumption of dilation invariance inorder to construct a theory which admits a solution that obeys the null energy condition at earlytimes but at late times crosses into a phase of NEC-violation. The theory of interest is a deformationof the Galilean Genesis Lagrangian, L = Z ( π ) e π ( ∂π ) + f Λ ( ∂π ) (cid:3) π + 1 J ( π ) f ( ∂π ) , (3)where the functions Z ( π ) and J ( π ) are constrained to allow a smooth interpolation between aNEC-satisfying phase at early times and a NEC-violating phase at late times. Specifically, at earlytimes ( π → π ∞ ) the cubic term is negligible, and the theory reduces to L early (cid:39) − f ∞ e π ∞ ( e π − π ∞ − ( ∂π ) + 1 J f ( ∂π ) (cid:0) − e − ( π − π ∞ ) (cid:1) = − f ∞ e π ∞ ( ∂φ ) + f J Λ ( ∂φ ) , (4)where the second line follows after a field redefinition to the (almost) canonically-normalized variable φ . The quartic term has the correct sign demanded by locality [72], hence the S-matrix of this theoryobeys the standard dispersion relations coming from analyticity.At late times ( π → ∞ ), on the other hand, Z and J both tend to constants, Z ( π ) → f (cid:29) Λ , J ( π ) → J ∼ O (1), such that (3) reduces to the Galilean Genesis action L late (cid:39) f e π ( ∂π ) + f Λ ( ∂π ) (cid:3) π + 1 J f ( ∂π ) . (5)This gives rise to the usual, genesis NEC-violating solution. For suitable values of J , perturbationsaround this solution are comfortably subluminal, as in [59]. Note that Z ( π ) has the correct signat early times, and the wrong ( i.e. , ghost-like) sign at late times. Nevertheless, the kinetic term offluctuations around the time-dependent interpolating solution is healthy during the entire evolution.Of course, the presence of arbitrary functions in the Lagrangian makes it unlikely that the theoryis radiatively stable. However, we will argue that quantum effects in the theory are under controlboth at early and late times. We imagine that, given this, the existence of a healthy interpolatingsolution is not extremely sensitive to quantum corrections, even though the functions Z and J ,and hence the explicit form of the solution itself, might be. In this sense, our explicit constructionis designed to be a proof-of-principle.The paper is organized as follows. In Sec. 2 we briefly review earlier attempts to violate the NEC. InSec. 3 we review Rubakov’s argument which forbids, in dilation invariant theories, smooth solutions Although [65] focused on solutions which interpolate in a radial direction (a ‘bubble’ of NEC violation), theargument applies equally well to interpolation in the temporal direction. It was recently argued that including a matter component can reintroduce superluminality in some part of thecosmological phase space [73].
In this Section, we give a brief overview of the different theories that can violate the NEC, high-lighting their successes and failures. For a more comprehensive review of the null energy conditionand attempts to violate it, see [74]. In Table 1 we provide a scorecard for the different theories.A natural place to search for matter which can violate the NEC is in the context of scalar fieldtheories, since scalars can develop nontrivial background profiles that preserve homogeneity andisotropy. • Consider a non-linear sigma model with dynamical variables φ I : R , → M , where M is an arbitrary, N -dimensional real target space. At 2-derivative order,the action is given by S = (cid:90) d x (cid:18) − G IJ ( φ ) ∂ µ φ I ∂ µ φ J − V ( φ I φ I ) (cid:19) , (6)where G IJ is the target-space metric. The stress-energy tensor for this field is readily com-puted, and the quantity relevant for the NEC is T µν n µ n ν = G IJ ( φ ) n µ n ν ∂ µ φ I ∂ ν φ J . (7)In the language of perfect fluids, focusing on time-dependent profiles, φ I ( t ), this becomes ρ + P = G IJ ( φ ) ˙ φ I ˙ φ J . (8)Now, the target-space metric G IJ can be diagonalized, since it is symmetric and invertible.Therefore, in order to violate the NEC ( ρ + P ≤ G IJ must have at least one negativeeigenvalue, that is, one of the φ I ’s must be a ghost. At the 2-derivative level, violating theNEC comes hand in hand with ghosts. • P ( X ) theories: The obvious generalization is to consider higher-derivative theories. In orderto avoid ghost instabilities, the equation of motion should remain 2 nd -order. A general classof such models is S = M (cid:90) d xP ( X ) , (9)where X ≡ − M ( ∂φ ) , and M is an arbitrary mass scale. The justification for consideringtheories of this type is effective field theory reasoning — we anticipate that at low enoughenergies, terms with more derivatives per field will be sub-leading. However, even in thesetheories, NEC violation generically introduces pathologies, albeit of a more subtle nature. Throughout, we use the mostly plus ( − , + , + , +) metric convention.
4o see this, note that the combination ρ + P , assuming φ = φ ( t ), is given by ρ + P = 2 XP ,X . (10)In order to violate the NEC, we therefore need P ,X <
0. Meanwhile, expanding (9) about thebackground φ = ¯ φ ( t ) + ϕ , the action for quadratic fluctuations is [67] S ϕ = 12 (cid:90) d x (cid:16) ( P, X +2 XP, XX ) ˙ ϕ − P, X ( (cid:126) ∇ ϕ ) (cid:17) . (11)A violation of the NEC ( P, X <
0) results in either gradient instabilities (wrong-sign spatialgradient term) or ghost instabilities (if we choose P, X +2 XP, XX < L ( φ I , ∂φ I ) ( i.e. , involving atmost one derivative per field), implies either the presence of ghost or gradient instabilities or superluminal propagation. • Ghost condensation:
This general theorem about instabilities in such a wide class oftheories would seem to preclude any sensible violations of the NEC. There is, however, arather compelling loophole to the general logic. The theorem of [63] relies heavily on thestandard organization of effective field theory, i.e. , the sub-dominance of terms of the form ∂ φ . There exist two well-studied situations where such terms can become important and,indeed, both lead to violations of the NEC free of the obvious pathologies.The first is ghost condensation [67]. This relies on an action of the P ( X ) form (9), but chosenso that there exists a solution with P, X = 0. Notice from (11) that this precisely corresponds tothe vanishing of the spatial gradient term in the quadratic Lagrangian about this background.This allows a higher-derivative term of the form ( ∇ ϕ ) to become important in the quadraticLagrangian without the effective field theory expansion breaking down. Since ρ + P = 0 onthe background, this acts as a vacuum energy contribution. Deforming the background as φ = ¯ φ ( t ) + π ( t ), one finds ρ + P ∼ ˙ π . (12)This is linear in π , and hence can have either sign. Violating the NEC once again will pushthe kinetic term of fluctuations slightly negative, but the dispersion relation is stabilized athigh k by the ( ∇ ϕ ) term [53]. The no-go theorem of [63] is thus evaded by relying onhigher-derivative spatial gradient terms.The main drawback of the ghost condensate is the absence of a Lorentz-invariant vacuum.Indeed, from (11) the absence of ghosts about the condensate P ,X = 0 solution requires P ,XX > i.e. , the condensate is at a minimum of P ( X ). As a result, the theory cannot beconnected to a Lorentz-invariant vacuum ( P ,X | X =0 >
0) without encountering pathologies inbetween. The theory is only well-defined in the neighborhood of the ghost condensate point.A NEC-violating ghost condensate phase has been used in alternative cosmological models,including a universe starting from an asymptotically static past [53], the New EkpyroticUniverse [23, 26], and the matter-bounce scenario [39]. • Galileons:
A second class of theories which can violate the NEC without instabilities is givenby the conformal galileons [68, 76]. These are conformally-invariant scalar field theories with For another construction which violates the NEC based on Kinetic Gravity Braiding [62], a cousin of the galileons,see [77]. L = f e π ( ∂π ) + f Λ (cid:3) π ( ∂π ) + f ( ∂π ) . (13)Each term is manifestly dilation invariant. The relative 1 / (cid:3) π ( ∂π ) and ( ∂π ) ensures full conformal invariance. Choosing the kinetic term to have the wrongsign, as in (13), the theory admits a time-dependent solution e π = 1 H ( − t ) ; H = 23 Λ f , (15)where −∞ < t <
0. For consistency of the effective field theory, the scale H should lie belowthe strong coupling scale Λ, which requires f (cid:29) Λ . (16)This background spontaneously breaks the original SO(4 ,
2) symmetry down to its SO(4 , ρ + P = − f H t . Perturbationsaround this solution are stable, and propagate exactly luminally by SO(4 ,
1) invariance. How-ever, the sound speed can be pushed to superluminal values on slight deformations of thisbackground. A cure to this pathology [59] is to reduce the symmetry by detuning the relativecoefficient of the cubic and quartic terms L = f e π ( ∂π ) + f Λ (cid:3) π ( ∂π ) + f (1 + α )( ∂π ) , (17)where α is a constant. For α (cid:54) = 0, this explicitly breaks the special conformal symmetry,leaving dilation and Poincar´e transformations as the only symmetries (which convenientlyclose to form a subgroup). This still allows a 1 /t background of the form (15), with H =
23 1(1+ α ) Λ f depending on α . For − < α <
3, this background violates the NEC and has stableperturbations. As a result of the fewer residual symmetries, perturbations propagate with asound speed different from unity: c s = 3 − α α ) . (18)This is subluminal for α >
0. In other words, for the range0 < α < , (19)the system violates the NEC, is stable against small perturbations, and these perturbationspropagate at subluminal speeds. Moreover, the theory is stable against quantum corrections.The main drawback of the galileon NEC violation is — just like the ghost condensate — theabsence of a Lorentz-invariant vacuum. Indeed, the existence and stability of a 1 /t back-ground requires a wrong-sign kinetic term, as in (13) and (17). As shown in [59], including Under the dilation and conformal symmetries, the field π transforms as: δ D π = − − x µ ∂ µ π ; δ K µ π = − x µ − (2 x µ x ν ∂ ν − x ∂ µ ) π . (14) It was shown in [73] that such deformations must break homogeneity/isotropy. ρ + P for the 1 /t solution, and it must have the wrong sign to violate NEC. An im-provement over the ghost condensate, however, is that perturbations are stable on all scales,whereas perturbations of the ghost condensate in the NEC-violating phase are unstable onlarge scales (but are stabilized on small scales, thanks to the higher-derivative contributionto the dispersion relation).A NEC-violating galileon phase is the hallmark of the Galilean Genesis scenario [54, 59],in which the universe expands from an asymptotically static past. Because of the residualdilation symmetry, nearly massless fields acquire a scale invariant spectrum. The SO(4 , → SO(4 ,
1) spontaneous breaking is also used in the NEC-satisfying rolling scenario of [40, 41, 48].More generally, this symmetry breaking pattern arises whenever a number of scalar operators O I with weight ∆ I in a conformal field theory acquire a time-dependent profile O I ( t ) ∼ ( − t ) − ∆ I . The general effective action was constructed in [49] utilizing the coset construction,and the consistency relations were derived in [78]. • DBI Galileons:
An alternative way to avoid superluminality while preserving the fullSO(4 ,
2) symmetry is to consider the DBI conformal galileons [66]. These are the “rela-tivistic” extension of the ordinary conformal galileons, and describe the motion of a 3-branein an AdS geometry. The DBI conformal galileon action is a sum of five geometric invariants,with 5 free coefficients c , . . . , c : L = c L + c L + c L + c L + c L , (20)where the L i ’s are built out of the induced metric¯ g µν = G AB ∂ µ X A ∂ ν X B = φ (cid:18) η µν + ∂ µ φ∂ ν φφ (cid:19) , (21)the Ricci tensor ¯ R µν and scalar ¯ R , and the extrinsic curvature tensor K µν = γφ (cid:18) η µν − ∂ µ ∂ ν φφ + 3 ∂ µ φ∂ ν φφ (cid:19) . (22)Each L i is invariant up to a total derivative under SO(4 ,
2) transformations, inherited from theisometries of AdS . The relevant terms come from considering brane Lovelock invariants [79]7nd the boundary terms associated to bulk Lovelock invariants: L = − φ ; L = −√− ¯ g = − φ γ ; L = √− ¯ gK = − φ + φ [Φ] + γ φ (cid:16) − [ φ ] + 2 φ (cid:17) ; L = −√− ¯ g ¯ R = 12 φ γ + γφ (cid:16) [Φ ] − (cid:0) [Φ] − φ (cid:1) (cid:0) [Φ] − φ (cid:1) (cid:17) + 2 γ φ (cid:16) − [ φ ] + [ φ ] (cid:0) [Φ] − φ (cid:1) − φ + 6 φ (cid:17) ; L = 32 √− ¯ g (cid:18) − K K µν K − K µν − (cid:18) ¯ R µν −
12 ¯ R ¯ g µν (cid:19) K µν (cid:19) = 54 φ − φ [Φ] + γ φ (cid:18) φ ] φ + 2[Φ ] − ][Φ] + 12[Φ ] φ + [Φ] − φ + 42[Φ] φ − φ (cid:19) + 3 γ φ (cid:18) − φ ] + 2[ φ ] (cid:0) [Φ] − φ (cid:1) + [ φ ] (cid:0) [Φ ] − [Φ] + 8[Φ] φ − φ (cid:1) + 2 φ (cid:0) [Φ] − [Φ ] (cid:1) − φ + 12 φ (cid:19) , where γ ≡ / (cid:112) ∂φ ) /φ is the Lorentz factor for the brane motion, L measures theproper 5-volume between the brane and some fixed reference brane L is the world-volumeaction [80], i.e. , the brane tension, and the higher-order terms L , L and L are variousfunctions of curvature. Moreover, Φ denotes the matrix of second derivatives ∂ µ ∂ ν φ , [Φ n ] ≡ Tr(Φ n ), and [ φ n ] ≡ ∂φ · Φ n − · ∂φ , with indices raised by η µν . The motivation for consideringLovelock terms is that they lead to second-order equations of motion for the scalar field φ [66].For suitable choices of the coefficients c , . . . , c , the theory admits a 1 /t solution of theform (15), which violates the NEC in a stable manner [61]. This was dubbed the DBI Genesis phase in [61]. Analogous to DBI inflation [81], the sound speed of fluctuations for relativisticbrane motion γ (cid:29) γ .More importantly, the theory also admits a stable, Poincar´e-invariant vacuum. As such, DBIGenesis is the first example of a theory possessing both stable NEC-violating and stablePoincar´e-invariant vacua. In [61] it was shown that the 2 → host condensate Galilean Genesis DBI Genesis This theory (cid:24)(cid:24)(cid:24) NEC vacuum (cid:51) (cid:51) (cid:51) (cid:51)
No ghosts (cid:51) (cid:51) (cid:51) (cid:51)
Sub-luminality (cid:51) (cid:51) (cid:51) (cid:51)
Poincar´e vacuum (cid:55) (cid:55) (cid:51) (cid:51)
No ghosts (cid:51) (cid:51)
S-Matrix analyticity (2 → (cid:51) (cid:51) Sub-luminality (cid:55) (cid:51)
Interpolating solution (cid:55) (cid:51)
Radiative stability (cid:51) (cid:51) (cid:51) (cid:55)
BH Thermodynamics (cid:55) ? ? ?
Table 1:
Checklist of properties of various theories which possess null energy condition-violating solutions. condensate violation of the NEC allows for the formation of perpertuum mobile [69]. The storyis potentially more subtle for DBI galileons, thanks to the non-minimal terms required for theircovariantization [66, 83]. This is currently under investigation [84].Ideally, one would like to be able to start from the Poincar´e-invariant vacuum and evolve smoothlyinto the NEC-violating phase. As pointed out recently [65], however, this is impossible in anysingle-field theory with dilation invariance. This is particularly constraining because many of theattempts to violate the NEC utilize dilation-invariant theories (for example the Galilean Genesisscenarios and the DBI conformal galileons). The argument, reviewed below, shows that any solutionthat attempts to interpolate between the two vacua inevitably hits a strong coupling point. Oneway out is to invoke multiple scalar fields. Another way out, which we will explore here, is tobreak the dilation symmetry explicitly. In doing so, we will be able to construct a theory with thefollowing properties: • A Poincar´e-invariant vacuum with stable and sub-luminal fluctuations about this vacuum. • A solution which interpolates between a non-NEC-violating phase and a phase of NEC vio-lation with stability and sub-luminality for perturbations about this solution.
In this Section we review the no-go argument of Rubakov [65], which forbids the existence of a well-behaved solution interpolating between dilation-invariant vacua. The argument is very general andapplies to any single scalar field theory that enjoys (at the classical level) dilation invariance, andadmits both a Poincar´e invariant solution and a dilation-preserving, NEC-violating background.First note that conservation of the energy-momentum tensor is equivalent to the equation of motionvia ∂ µ T µν = − δSδπ ∂ ν π , (23)where δS/δπ is the Euler–Lagrange derivative. Specializing to π = π ( t ), it follows the equation of For a contrary viewpoint, see [82]. Z ( e − π ˙ π ) e − π ˙ π Figure 1:
In a dilation invariant theory, we must have Z = 0 at both e − π ˙ π = 0 and e − π ˙ π = Y , as wellas Z (cid:48) > Z (cid:48) <
0, as is clear from the plot. motion is equivalent to energy conservation:˙ ρ = − ˙ π δSδπ . (24)Now we assume that the equation of motion is second-order, that is, δS/δπ contains at most ¨ π butno higher-derivatives. It then follows that ρ must be a function only of π and ˙ π , for otherwise˙ ρ would contribute higher-derivative terms in (24). Since the theory is dilation invariant, we candeduce the form of ρ : ρ = e π Z (cid:0) e − π ˙ π (cid:1) , (25)where Z is a theory-dependent function.If the theory admits a Poincar´e-invariant solution, π = constant, it will have vanishing energydensity: Z (0) = 0 . (26)Additionally, if the theory admits a NEC-violating background which preserves homogeneity andisotropy, then π can only depend on time. If this background is also dilation invariant, then itmust take the form e π ∼ t − , and hence e − π ˙¯ π ≡ Y = constant on this solution. Moreover, theassumed symmetries imply ρ = βt − on the time-dependent solution, while energy conservationrequires ˙ ρ = 0, and thus β = 0. It follows that Z ( Y ) = 0 . (27)In other words, the energy density vanishes on any background that preserves homogeneity, isotropyand dilation symmetry. This of course includes the Poincar´e-invariant vacuum and (by assumption)the NEC-violating background.Next consider the stability of these solutions. We can expand (24) about some time-dependentbackground, π = ¯ π ( t ) + ϕ , and use the form (25) for ρ to derive an equation of motion for ϕ . Forthe diagnosis of ghost instabilities, we only explicitly need the ¨ ϕ term: − e π ¯ Z (cid:48) ¨ ϕ + · · · = 0 . (28) Note that violating this assumption would lead to Ostrogradski-type instabilities [85, 86]. Under a finite dilation, x µ (cid:55)→ λx µ , the field π transforms as π ( x ) (cid:55)→ π ( λx ) + log λ . One can then check that (25)is the most general object depending only on π and ˙ π invariant under this symmetry. L = e π ¯ Z (cid:48) ˙ ϕ + · · · . In order for ϕ to be healthy,we must have ¯ Z (cid:48) > . (29)The problem is now clear: we have two backgrounds, each with Z = 0. In order for them toboth be healthy, we must have Z (cid:48) > Z (cid:48) < Z = 0, which correspondsto strong coupling. It is therefore impossible to connect the two backgrounds with a solution whichis perturbative. See Fig. 1 for a graphical representation of this result.This no-go argument is very general, but we can get some inspiration for how to avoid it byexamining its assumptions. The most natural ones to consider breaking are the assumption of asingle degree of freedom and that of dilation invariance. Indeed, Rubakov considers a model whichintroduces additional degrees of freedom to construct an interpolating solution [65]. Here, we willfocus on theories that are not dilation invariant. To circumvent the no-go argument of Sec. 3, we stick to a single-field theory but relax the assump-tion of dilation invariance. We consider a deformation of the conformal galileon lagranagian (13)(used in Galilean Genesis [54, 68]) by introducing functions Z ( π ) , J ( π ) which explicitly break scaleinvariance: L = Z ( π ) e π ( ∂π ) + f Λ ( ∂π ) (cid:3) π + 1 J ( π ) f ( ∂π ) . (30)Our goal is to find suitable functional forms for Z ( π ) and J ( π ) such that the theory admits asmooth solution which is NEC-satisfying at early times ( t (cid:28) t ∗ ), and NEC-violating at late times( t (cid:29) t ∗ ). The transition time will be denoted by t ∗ . To achieve NEC violation with strictly subluminal propagation of perturbations at late times ( t (cid:29) t ∗ ), the theory should approximate the form (17), used in subluminal genesis [59]. This requires Z ( π ) → f ; J ( π ) → J for t (cid:29) t ∗ , (31)where f (cid:29) Λ and J is an O (1) constant. Thus, the theory reduces at late times to L late (cid:39) f e π ( ∂π ) + f Λ ( ∂π ) (cid:3) π + 1 J f ( ∂π ) . (32)Comparison with (17) gives the translation J = 11 + α , (33)11ence we anticipate that we will need J ∼ < O (1) to have subluminality [59]. At late times, thesolution should therefore asymptote to the Genesis background e π = 1 H ( − t ) ; H = 2 J f for t (cid:29) t ∗ . (34)The energy scale of this solution is H . We demand that it lie below the strong coupling scale ofthe effective theory, H (cid:28) Λ, which will be the case if f (cid:29) Λ . (35)The background (34) is a solution on flat, Minkowski space. With gravity turned on, it remainsan approximate solution at early enough times in the Genesis phase. Gravity eventually becomesimportant at a time t end , which will be computed in Sec. 5. At early times ( t (cid:28) t ∗ ), the solution should asymptote to a constant field profile: π (cid:39) π ∞ for t (cid:28) t ∗ . (36)In order for this constant background to be ghost-free, the sign of the kinetic term should be theusual (negative) one: Z ( π ) < t (cid:28) t ∗ . (37)We will see that this gives rise to a NEC-satisfying phase, with ρ ∼ P . In this regime, clearlygravity cannot be ignored arbitrarily far in the past, since the universe must emerge from a bigbang singularity. We will come back to this point in Sec. 5 and show that the time t beg wheregravity becomes important is parametrically larger in magnitude than t ∗ . In other words, there isa parametrically large window t beg (cid:28) t (cid:28) t ∗ within which gravity is negligible and the early-timeexpressions above hold. In particular, π can be approximated as constant over this regime, in thesense that it varies slowly compared to the Hubble parameter at the transition.From (31) and (37), note that Z ( π ) has the correct sign at early times, and the wrong ( i.e. ,ghost-like) sign at late times (as required for the Genesis solution). Nevertheless, we will see thatthe kinetic term of fluctuations around the time-dependent solution is healthy during the entireevolution. This does imply, however, that stable, Lorentz-invariant vacua only exist for a finiterange in field space. We engineer the desired Z ( π ) and J ( π ) by demanding that they give rise to a suitable time-dependent background solution, which interpolates between π (cid:39) π ∞ at early times and e π ∼ /t at late times. A simple ansatz for the background which satisfies these asymptotic conditions is e ¯ π ( t ) = e π ∞ (cid:18) t ∗ t (cid:19) , (38)where t ∗ sets the transition time. 12ssuming spatial homogeneity, the equation of motion for π = ¯ π ( t ) following from (30) is¨¯ π (cid:18) −Z (¯ π ) e π + 3 J (¯ π ) f Λ ˙¯ π (cid:19) = ˙¯ π (cid:18)(cid:0) Z (¯ π ) + Z (cid:48) (¯ π ) (cid:1) e π + 3 J (cid:48) (¯ π )2 J (¯ π ) f Λ ˙¯ π (cid:19) . (39)This admits a first integral of motion enforcing energy conservation: ρ = −Z (¯ π ) e π ˙¯ π + 32 J (¯ π ) f Λ ˙¯ π = −Z ( t ) e π ∞ t ∗ t + 32 J ( t ) f Λ t ∗ t ( t + t ∗ ) = constant , (40)where in the second line we have substituted in the background solution (38). At early times( t (cid:28) t ∗ ), the two contributions scale differently: ∼ t − for the first term; ∼ t − for the second. Thesimplest option is for each term to be separately constant, from which we can deduce the scaling Z ( t ) ≈ t and J ( t ) ≈ t − for t (cid:28) t ∗ . A nice choice for J ( t ) with this property (and satisfying J (cid:39) J for t (cid:28) t ∗ ) is J ( t ) = J (1 + tt ∗ ) . (41)Equivalently, using (38), J ( π ) = (cid:16) − e − ( π − π ∞ ) (cid:17) J . (42)Substituting J ( t ) into the integrated equation of motion (40) yields Z ( t ) = − (cid:0) f ∞ + f (cid:1) t t ∗ + (cid:18) tt ∗ (cid:19) f , (43)where f ∞ , introduced for reasons that will soon become obvious, is related to the energy densityby ρ = 32 t ∗ (cid:18) f ∞ f (cid:19) f J Λ . (44)Moreover, we can obtain an expression for π ∞ and the transition time: e π ∞ = (cid:114) f J Λ | t ∗ | . (45)In terms of π , the function Z can be expressed as Z ( π ) = f ( e π − π ∞ − (cid:32) e π − π ∞ ) − (cid:18) f ∞ f (cid:19) (cid:33) . (46)Hence we have 5 parameters defining the theory: f , f ∞ , Λ, J and π ∞ . The transition time t ∗ is not a free parameter, as it is set by the other parameters in the Lagrangian. By construction,the Lagrangian (30) with the functions (42) and (46) admits the interpolating solution (38) as asolution to its equations of motion. 13 .4 Early times revisited With the expressions above for Z ( π ) and J ( π ), we can investigate the action at large, constantfield values, where it takes the approximate form (note that this limit is essentially the early timelimit on the solution (38)): L early (cid:39) − f ∞ e π ∞ ( e π − π ∞ − ( ∂π ) + 1 J f ( ∂π ) (cid:0) − e − ( π − π ∞ ) (cid:1) , (47)It is convenient to define the almost-canonically-normalized variable, φ = 11 − e − ( π − π ∞ ) . (48)The virtue of this redefinition is that in terms of φ , the background solution (38) reduces to a linearform ¯ φ ( t ) = 1 + tt ∗ . (49)Another benefit is that the functions Z ( π ) and J ( π ) simplify to Z ( φ ) = f φ − (cid:0) f + f ∞ (cid:1) ( φ − ; J ( φ ) = J φ . (50)Furthermore, the early-time action in terms of φ reduces to L early (cid:39) − f ∞ e π ∞ ( ∂φ ) + f J Λ ( ∂φ ) . (51)The kinetic term is healthy, as it should be, hence the theory admits Poincar´e-invariant solutions.Further, the quartic term is manifestly positive : this ensures both a lack of superluminality aboutthese Poincar´e-invariant vacua and that the simplest dispersion relations following from S-matrixanalyticity [72] are satisfied. The reduced symmetry of the action due to the presence of the arbitrary functions Z ( π ) and J ( π )makes it unlikely that the theory will be stable under quantum corrections. However, all is notlost. Recall that for large constant field values the action (30) can be cast as L early (cid:39) f ∞ (cid:18) − e π ∞ ( ∂φ ) + f f ∞ J Λ ( ∂φ ) (cid:19) . (52)In this way, f ∞ plays a role analogous to 1 / (cid:126) ; for sufficiently large f ∞ , quantum effects can bemade negligible and the theory will be radiatively stable [68]. Similarly, at late times (or, as π → ∞ ), the theory can be cast as L late (cid:39) f e π ( ∂π ) + f Λ ( ∂π ) (cid:3) π + 1 J f ( ∂π ) , (53) Another way of saying this is that terms radiatively generated in the Lagrangian (52) will be suppressed bypowers of f ∞ , and can be ignored for sufficiently large values of f ∞ . Z ( π ) and J ( π ) are stable at both ends of the evolution. Inbetween, most likely they will be greatly affected by quantum corrections. However, the fact thattheir asymptotic forms are preserved makes it plausible that a solution which interpolates betweenNEC-violating and NEC-satisfying regions will continue to exist. Although the detailed form of thesolution will surely be modified, we do not expect its stability properties to be greatly affected, aswe are able to satisfy the stability requirements for a wide range of parameters. In this sense theexplicit interpolating form for Z ( π ) and J ( π ) constructed above is a proof of principle. It is straightforward to calculate the stress-energy tensor for the Lagrangian (30), in the approxi-mation that the gravitational background is Minkowski space. The energy density is constant andhas already been given in (40) and (44). The pressure is given by P = −Z (¯ π ) e π ˙¯ π + 12 J (¯ π ) f Λ ˙¯ π − f Λ ˙¯ π ¨¯ π . (54)On the solution (38), at late times this reduces to P late (cid:39) − (cid:18) J + 2 (cid:19) f Λ t . (55)Since the late-time pressure grows as 1 /t while the energy density remains constant, the NEC willviolated at late times if P late <
0. This requires1 J > − . (56)This is the NEC-violating genesis phase.At early times, meanwhile, the pressure is constant and positive: P early (cid:39) t ∗ (cid:18) f ∞ f (cid:19) f J Λ . (57)Hence the early-time regime is NEC-satisfying.More generally, by combining (40) and (54) we see that the NEC is violated whenever ρ + P = 2 (cid:18) − Z ( π ) e π ˙ π + 1 J ( π ) f Λ ˙ π − f Λ ˙ π ¨ π (cid:19) < . (58)where we have dropped the bars for simplicity. This condition can be studied numerically. Forthis purpose, we will focus on “on-shell” solutions, that is, on profiles ¯ π ( t ) that are solutions tothe equation of motion (39). This allows us to rewrite ¨ π as a function of π and ˙ π . Moreover, it is As a check, translating to the α parameter of Subluminal Genesis via (33), this matches the pressure computedin [59]. EC-SatisfyingNEC-ViolatingNEC-Violating Φ Φ (cid:162) Figure 2:
NEC-satisfying and NEC-violating regions in the ( φ, φ (cid:48) ) phase space, for the parameter values f ∞ f = 10 and J = 0 .
75. The solution of interest, φ = 1 + t/t ∗ , corresponding to φ (cid:48) = 1, is plotted as a blackdashed line. It first obeys the NEC for a period of time, and then crosses into the NEC-violating regime. convenient to express the result in terms of the φ variable introduced in (48), since its backgroundevolution is particularly simple. The NEC-violating region in phase space corresponds to −
32 1 f J Z ( φ ) φ + (cid:18) J ( φ ) − φ + 1 (cid:19) φ (cid:48) − φ (cid:48) ( φ (cid:48) − − φ (cid:48) − (cid:16) f ∞ f (cid:17) (1 − φ − ) < , (59)where we have defined φ (cid:48) ≡ t ∗ ˙ φ . The result is plotted in the ( φ, φ (cid:48) ) plane in Fig. 2 for a fiducialchoice of parameters.All of the results up to this point have been derived under the approximation that gravitationalbackreaction can be neglected. We will now quantify the time interval over which this assumptionis justified. Consider first the early-time regime. Since pressure and energy density are comparable(and constant) in this epoch, gravitational backreaction can only be neglected for at most a Hubbletime H − = (cid:113) M /ρ . Our approximation is therefore justified for t (cid:29) t beg , where (ignoring O (1)coefficients) t beg t ∗ ∼ (cid:113) f ∞ f (cid:115) Λ f | t ∗ | M Pl . (60)For consistency, we must have | t beg | (cid:29) | t ∗ | , that is, | t ∗ | (cid:29) M Pl (cid:115) f ∞ f (cid:114) f Λ . (61)Determining the evolution before t beg , including gravity, would require a detailed calculation. Butsince the NEC is preserved, the answer is qualitatively simple: within a time of order t beg , theevolution must trace back to a big bang singularity.During the Genesis phase, on the other hand, the gravitational dynamics are dominated by the16igure 3: Timeline for the evolution. Our approximation of neglecting gravity is valid for the range t beg ≤ t ≤ t end . For t < t beg , the universe asymptotes to a big bang singularity (since the NEC is satisfiedin 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 theNEC-violating phase to a standard, radiation-dominated phase. pressure (55). Integrating M ˙ H (cid:39) − P , we have H late (cid:39) M | t | (1 + 2 J ) f J Λ , (62)corresponding to a time-dependent contribution to the energy density: ρ late = 112 M t (1 + 2 J ) (cid:18) f J Λ (cid:19) . (63)This dominates over the constant piece (44). Gravitational backreaction can be neglected as longas P late (cid:29) ρ late . This breaks down at a time t end obtained by setting P late ∼ ρ late : | t end | ∼ M Pl (cid:114) f Λ . (64)The condition f (cid:29) Λ mentioned in (35) ensures that | t end | (cid:29) M − . Moreover, the condition (61)automatically implies that | t end | (cid:28) | t ∗ | , which is obviously required for consistency.To summarize, our approximation of neglecting gravity is valid over the interval t beg (cid:28) t (cid:28) t end . (65)In order for the transition time to lie within this interval, t ∗ must satisfy the condition (61). Thetime-line for the entire evolution is sketched in Fig. 3. With expressions for P , ρ , t beg , and t end , it is possible to show how the scale factor may smoothlytransition from a decreasing phase to and increasing one. Recall that the Hubble parameter obeys(2): M ˙ H = −
12 ( ρ + P ) . (66)Combined with the expressions for ( ρ + P ) (58) and e π ( t ) (38), we obtain the following expressionfor ˙ H ( t ): ˙ H ( t ) = − f M J Λ t ∗ (cid:32) (cid:18) f ∞ f (cid:19) − (cid:18) t ∗ t (cid:19) − J t ∗ t tt ∗ + 1(1 + tt ∗ ) (cid:33) . (67)17 EC-Satisfying
NEC-Violating t beg t end t (cid:42) tH Figure 4:
The Hubble parameter is plotted for two values of H i with the fiducial parameters J = 0 . f ∞ /f = 10, Λ = 0 . t ∗ = −
1, and C = − . × − M Pl . The vertical dashed line marks the boundarybetween the NEC-satisfying and NEC-violating phases. As we would expect, ˙ H <
H > H i = 0 . M Pl ) always has H > t < t beg . The solid line (with H i = − . M Pl ) represents an initially contracting universe( H <
0) which enters the phase of NEC-violation and undergoes a cosmological bounce to an acceleratingphase (
H >
This expression is valid in the range t end (cid:28) t (cid:28) t beg . It may be integrated to give H in this range: H ( t ) = − f M J Λ t ∗ (cid:32) (cid:18) f ∞ f (cid:19) tt ∗ + t ∗ t + t ∗ t + 3 t ∗ t − tt ∗ − t t ∗ + J t ∗ t tt ∗ ) (cid:33) + C + H i (68) C and H i are both integration constants, but C is chosen to ensure that H ( t beg ) = H i . Thisis plotted over the range of validity in Fig. 4 for two different values of H i . The dashed linesolution represents an expanding universe that originated in a big bang, and the solid line is acontracting universe that might have originated from Minkowski space or a big bang singularity inthe asymptotic past. Although both solutions demonstrate the transition from ˙ H <
H >
H <
H > i.e. , a bounce.Notice that this bounce occurs before t end , our estimate for when gravitational back-reaction canno longer be ignored in the solution for π . This indicates that we can trust the existence of thebounce within our effective theory. We now turn to the study of perturbations around background solutions. Expanding the La-grangian (30) to quadratic order in perturbations ϕ = π − ¯ π , we find L quad = Z ϕ ( t ) ˙ ϕ − K ϕ ( t )( ∇ ϕ ) , (69)18 radient Instability Gradient Instability Φ Φ (cid:162) (a) Gradient Instability Superluminality Φ Φ (cid:162) (b) Superluminality Strong Coupling Φ Φ (cid:162) (c) Strong Coupling Figure 5:
The shaded regions represent parts of the ( φ, φ (cid:48) ) phase space where perturbations (a) suffer fromgradient instabilities; (b) propagate superluminally; (c) are strongly coupled. The parameter values are f ∞ f = 10 and J = 0 .
75. The solution of interest, φ = 1 + t/t ∗ , corresponding to φ (cid:48) = 1, is plotted as a blackdashed line. It avoids all pathological regions. where we have defined the functions Z ϕ ( t ) ≡ (cid:18) −Z (¯ π ) e π + 3 J (¯ π ) f Λ ˙¯ π (cid:19) ; K ϕ ( t ) ≡ (cid:18) −Z (¯ π ) e π + 2 f Λ ¨¯ π + 1 J (¯ π ) f Λ ˙¯ π (cid:19) . (70)The constraints on the quadratic theory are the following: • Absence of ghosts : To avoid ghosts, the kinetic term should be positive: Z ϕ >
0. It isstraightforward to show that will be the case if J > . (71)In particular, the NEC-violating condition (56) follows automatically.19 Absence of gradient instabilities : Similarly, the spatial gradient term should be positive: K ϕ >
0. Expressing this condition in terms of φ , and dropping the bars for simplicity, weobtain −
32 1 f J Z ( φ ) φ + (cid:18) φ −
1) + 1 J ( φ ) (cid:19) φ (cid:48) + 4 φ (cid:48) ( φ (cid:48) − − φ (cid:48) − (cid:16) f ∞ f (cid:17) (1 − φ − ) > , (72)where we recall that φ (cid:48) ≡ ˙ φt ∗ . • Subluminality : The final constraint at the quadratic level is to demand subluminal propa-gation: K ϕ / Z ϕ <
1. Assuming that both (71) and (72) are satisfied, subluminality follows bydefinition if the kinetic term is larger than the gradient term. It is straightforward to showthat this will be the case if (cid:18) J ( φ ) − φ + 1 (cid:19) φ (cid:48) − φ (cid:48) ( φ (cid:48) − − φ (cid:48) − (cid:16) f ∞ f (cid:17) (1 − φ − ) > . (73)In the genesis regime (corresponding to φ → φ (cid:48) → J : J < . (74)Beyond the quadratic theory, we should also check that the interactions are perturbative. Considerthe cubic vertex for the perturbations: L = f Λ ( ∂ϕ ) (cid:3) ϕ , (75)After canonical normalization of the kinetic term, ϕ c ≡ Z / ϕ ϕ , the cubic term becomes suppressedby the effective strong coupling scale Λ eff = Λ f Z / ϕ . (76)For consistency of the effective field theory, the characteristic frequency of the background, namely˙¯ π , should lie below this cutoff: ˙¯ π (cid:28) Λ eff . (77)In the Genesis phase, in particular, ˙¯ π (cid:39) /t sets the scale at which perturbations freeze out.Hence (77) is necessary to consistently describe the generation of perturbations within the effectivetheory. A straightforward calculation shows that this condition implies:2 ¯ φ (cid:48) (cid:18) J Λ3 f ¯ φ − (cid:19) (cid:28) (cid:18) f ∞ f (cid:19) (cid:0) − ¯ φ − (cid:1) − . (78)Note that the left-hand side is negative-definite within the range 0 < J < J Λ f (cid:28) , which is anotherway to confirm the condition f (cid:29) Λ mentioned in (35).In summary, the allowed range of J values is0 < J < . (79)20 EC-SatisfyingNEC-Violating
PathologyPathology Φ Φ (cid:162) Figure 6:
Phase portrait with all constraints overlaid, again for the fiducial choice of parameters f ∞ f = 10and J = 0 .
75. The shaded region represents the union of all pathological regions shown in Fig. 5. Thegreen long-dashed line separates the NEC-satisfying and NEC-violating regions. The black short-dashed linecorresponds to the background solution of interest, given by (49). The solid lines represent other backgroundsolutions (with different initial conditions).
The remaining constraints — no gradient instabilities (72), subluminality (74), and weak cou-pling (78) — are plotted in the ( φ, φ (cid:48) ) phase space in Fig. 5 for a fiducial choice of parameters.These constraints are overlaid in Fig. 6 with a range of solutions to the equation of motion. Onthese plots, the background solution (49) of interest corresponds to φ (cid:48) = 1. Other backgroundsolutions, corresponding to different initial conditions, are also plotted as solid lines. This showsthat there is a wide range of trajectories that interpolate between a constant field profile at earlytimes and the Genesis solutions at late times, while avoiding the pathological region at all times.Furthermore, it is clear that the background solution φ (cid:48) = 1 is an attractor at late times. It has proven surprisingly difficult to violate the null energy condition with a well-behaved rel-ativistic quantum field theory. In the simplest attempts, violating the NEC generally introducesghost instabilities, gradient instabilities, superluminality, or absence of a Lorentz-invariant vacuum.Progress has been made in avoiding some of these shortcomings, but a fully satisfactory exampleremains to be found. The null energy condition appears to be connected to some fundamentalphysics principles, such as black hole thermodynamics and the (non)-existence of cosmologicalbounces. Therefore, if it turns out that violating the NEC is impossible, pinpointing which of theaforementioned pathologies is the real roadblock will tell us something fundamental.The recently-proposed DBI Genesis scenario is the first example of a theory admitting both aPoincar´e-invariant vacuum and NEC-violating solutions. As argued by Rubakov, however, thesetwo backgrounds lie on different branches of solutions and cannot be connected by a smooth solutionwithout strong coupling occurring. This is an immediate consequence of dilation invariance.Here, we have abandoned dilation symmetry in order to circumvent Rubakov’s no-go argument. We21ave constructed a theory which admits a time-dependent solution that smoothly interpolates be-tween a NEC-satisfying phase at early times and a NEC-violating phase at late times. There existsa wide range of parameters for which perturbations around the background are stable, comfortablysubluminal and weakly-coupled at all times.The main drawback of the construction is the presence of suitably-engineered interpolating functionsin the action. It is highly unlikely that the detailed form of these functions will be preserved byquantum corrections. However, we argued that their asymptotic forms both in the past and inthe future are radiatively stable. Moreover, our analysis did not depend sensitively on the detailsof the interpolation. Therefore, all we need is for the quantum-corrected action to still allow aninterpolation between NEC-satisfying and NEC-violating solutions. We leave a detailed analysis ofradiative stability to future work.Another drawback of the explicit example presented here is that the kinetic term flips sign aswe adiabatically vary φ . It is healthy at early times, consistent with Poincar´e invariance, butbecomes ghost-like at late times, which is necessary to obtain a NEC-violating solution with thecubic Galilean Genesis action. Of course, as mentioned earlier, perturbations around the time-dependent background are always healthy. However, it would be aesthetically desirable if theperturbations around φ = constant backgrounds were also healthy for all field values of interest.This should be achievable by deforming the DBI Genesis Lagrangian, since this theory preciselysatisfies this property while allowing a NEC-violating background. We plan to study the DBIGenesis generalization in the future. It is also possible that the DBI extension will alleviate thequantum stability issues discussed above. Acknowledgements:
We would like to thank Lasha Berezhiani and Kurt Hinterbichler for usefuldiscussions. This work is supported in part by NASA ATP grant NNX11AI95G, NSF CAREERAward PHY-1145525 (B.E. and J.K.); the Kavli Institute for Cosmological Physics at the Universityof Chicago through grant NSF PHY-1125897 and an endowment from the Kavli Foundation andits founder Fred Kavli and by the Robert R. McCormick Postdoctoral Fellowship (A.J.).
References [1] J. M. Bardeen, B. Carter, and S. Hawking, “The Four laws of black hole mechanics,”Commun.Math.Phys. (1973) 161–170.[2] M. Gasperini and G. Veneziano, “Pre - big bang in string cosmology,” Astropart.Phys. (1993) 317–339, arXiv:hep-th/9211021 [hep-th] .[3] M. Gasperini and G. Veneziano, “The Pre - big bang scenario in string cosmology,”Phys.Rept. (2003) 1–212, arXiv:hep-th/0207130 [hep-th] .[4] M. Gasperini and G. Veneziano, “String Theory and Pre-big bang Cosmology,” arXiv:hep-th/0703055 [hep-th] .[5] J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, “The Ekpyrotic universe: Collidingbranes and the origin of the hot big bang,” Phys.Rev. D64 (2001) 123522, arXiv:hep-th/0103239 [hep-th] . 226] R. Y. Donagi, J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, “Visible branes withnegative tension in heterotic M theory,” JHEP (2001) 041, arXiv:hep-th/0105199[hep-th] .[7] J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Steinhardt, and N. Turok, “From big crunch to bigbang,” Phys.Rev.
D65 (2002) 086007, arXiv:hep-th/0108187 [hep-th] .[8] J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, “Density perturbations in theekpyrotic scenario,” Phys.Rev.
D66 (2002) 046005, arXiv:hep-th/0109050 [hep-th] .[9] D. H. Lyth, “The Primordial curvature perturbation in the ekpyrotic universe,” Phys.Lett.
B524 (2002) 1–4, arXiv:hep-ph/0106153 [hep-ph] .[10] R. Brandenberger and F. Finelli, “On the spectrum of fluctuations in an effective field theoryof the Ekpyrotic universe,” JHEP (2001) 056, arXiv:hep-th/0109004 [hep-th] .[11] P. J. Steinhardt and N. Turok, “Cosmic evolution in a cyclic universe,” Phys.Rev.
D65 (2002) 126003, arXiv:hep-th/0111098 [hep-th] .[12] A. Notari and A. Riotto, “Isocurvature perturbations in the ekpyrotic universe,” Nucl.Phys.
B644 (2002) 371–382, arXiv:hep-th/0205019 [hep-th] .[13] F. Finelli, “Assisted contraction,” Phys.Lett.
B545 (2002) 1–7, arXiv:hep-th/0206112[hep-th] .[14] S. Tsujikawa, R. Brandenberger, and F. Finelli, “On the construction of nonsingular pre - bigbang and ekpyrotic cosmologies and the resulting density perturbations,” Phys.Rev.
D66 (2002) 083513, arXiv:hep-th/0207228 [hep-th] .[15] S. Gratton, J. Khoury, P. J. Steinhardt, and N. Turok, “Conditions for generatingscale-invariant density perturbations,” Phys.Rev.
D69 (2004) 103505, arXiv:astro-ph/0301395 [astro-ph] .[16] A. J. Tolley, N. Turok, and P. J. Steinhardt, “Cosmological perturbations in a big crunch /big bang space-time,” Phys.Rev.
D69 (2004) 106005, arXiv:hep-th/0306109 [hep-th] .[17] B. Craps and B. A. Ovrut, “Global fluctuation spectra in big crunch / big bang stringvacua,” Phys.Rev.
D69 (2004) 066001, arXiv:hep-th/0308057 [hep-th] .[18] J. Khoury, P. J. Steinhardt, and N. Turok, “Inflation versus cyclic predictions for spectraltilt,” Phys.Rev.Lett. (2003) 161301, arXiv:astro-ph/0302012 [astro-ph] .[19] J. Khoury, P. J. Steinhardt, and N. Turok, “Designing cyclic universe models,”Phys.Rev.Lett. (2004) 031302, arXiv:hep-th/0307132 [hep-th] .[20] J. Khoury, “A Briefing on the ekpyrotic / cyclic universe,” arXiv:astro-ph/0401579[astro-ph] .[21] P. Creminelli, A. Nicolis, and M. Zaldarriaga, “Perturbations in bouncing cosmologies:Dynamical attractor versus scale invariance,” Phys.Rev. D71 (2005) 063505, arXiv:hep-th/0411270 [hep-th] . 2322] J.-L. Lehners, P. McFadden, N. Turok, and P. J. Steinhardt, “Generating ekpyrotic curvatureperturbations before the big bang,” Phys.Rev.
D76 (2007) 103501, arXiv:hep-th/0702153[HEP-TH] .[23] E. I. Buchbinder, J. Khoury, and B. A. Ovrut, “New Ekpyrotic cosmology,” Phys.Rev.
D76 (2007) 123503, arXiv:hep-th/0702154 [hep-th] .[24] E. I. Buchbinder, J. Khoury, and B. A. Ovrut, “On the initial conditions in new ekpyroticcosmology,” JHEP (2007) 076, arXiv:0706.3903 [hep-th] .[25] E. I. Buchbinder, J. Khoury, and B. A. Ovrut, “Non-Gaussianities in new ekpyroticcosmology,” Phys.Rev.Lett. (2008) 171302, arXiv:0710.5172 [hep-th] .[26] P. Creminelli and L. Senatore, “A Smooth bouncing cosmology with scale invariantspectrum,” JCAP (2007) 010, arXiv:hep-th/0702165 [hep-th] .[27] K. Koyama and D. Wands, “Ekpyrotic collapse with multiple fields,” JCAP (2007) 008, arXiv:hep-th/0703040 [HEP-TH] .[28] K. Koyama, S. Mizuno, and D. Wands, “Curvature perturbations from ekpyrotic collapsewith multiple fields,” Class.Quant.Grav. (2007) 3919–3932, arXiv:0704.1152 [hep-th] .[29] K. Koyama, S. Mizuno, F. Vernizzi, and D. Wands, “Non-Gaussianities from ekpyroticcollapse with multiple fields,” JCAP (2007) 024, arXiv:0708.4321 [hep-th] .[30] J.-L. Lehners and P. J. Steinhardt, “Non-Gaussian density fluctuations from entropicallygenerated curvature perturbations in Ekpyrotic models,” Phys.Rev. D77 (2008) 063533, arXiv:0712.3779 [hep-th] .[31] J.-L. Lehners and P. J. Steinhardt, “Intuitive understanding of non-gaussianity in ekpyroticand cyclic models,” Phys.Rev.
D78 (2008) 023506, arXiv:0804.1293 [hep-th] .[32] J.-L. Lehners and P. J. Steinhardt, “Non-Gaussianity Generated by the Entropic Mechanismin Bouncing Cosmologies Made Simple,” Phys.Rev.
D80 (2009) 103520, arXiv:0909.2558[hep-th] .[33] Y.-F. Cai, D. A. Easson, and R. Brandenberger, “Towards a Nonsingular BouncingCosmology,” JCAP (2012) 020, arXiv:1206.2382 [hep-th] .[34] T. Qiu, J. Evslin, Y.-F. Cai, M. Li, and X. Zhang, “Bouncing Galileon Cosmologies,” JCAP (2011) 036, arXiv:1108.0593 [hep-th] .[35] J. Khoury and P. J. Steinhardt, “Adiabatic Ekpyrosis: Scale-Invariant CurvaturePerturbations from a Single Scalar Field in a Contracting Universe,” Phys.Rev.Lett. (2010) 091301, arXiv:0910.2230 [hep-th] .[36] J. Khoury and P. J. Steinhardt, “Generating Scale-Invariant Perturbations fromRapidly-Evolving Equation of State,” Phys.Rev.
D83 (2011) 123502, arXiv:1101.3548[hep-th] .[37] A. Joyce and J. Khoury, “Scale Invariance via a Phase of Slow Expansion,” Phys.Rev.
D84 (2011) 023508, arXiv:1104.4347 [hep-th] .2438] J. Fonseca and D. Wands, “Tilted Ekpyrosis,” Phys.Rev.
D84 (2011) 101303, arXiv:1109.0448 [hep-th] .[39] C. Lin, R. H. Brandenberger, and L. Perreault Levasseur, “A Matter Bounce By Means ofGhost Condensation,” JCAP (2011) 019, arXiv:1007.2654 [hep-th] .[40] B. Craps, T. Hertog, and N. Turok, “On the Quantum Resolution of CosmologicalSingularities using AdS/CFT,” Phys.Rev.
D86 (2012) 043513, arXiv:0712.4180 [hep-th] .[41] V. Rubakov, “Harrison-Zeldovich spectrum from conformal invariance,” JCAP (2009)030, arXiv:0906.3693 [hep-th] .[42] M. Osipov and V. Rubakov, “Scalar tilt from broken conformal invariance,” JETP Lett. (2011) 52–55, arXiv:1007.3417 [hep-th] .[43] M. Libanov and V. Rubakov, “Cosmological density perturbations from conformal scalarfield: infrared properties and statistical anisotropy,” JCAP (2010) 045, arXiv:1007.4949 [hep-th] .[44] M. Libanov, S. Mironov, and V. Rubakov, “Properties of scalar perturbations generated byconformal scalar field,” Prog.Theor.Phys.Suppl. (2011) 120–134, arXiv:1012.5737[hep-th] .[45] M. Libanov, S. Ramazanov, and V. Rubakov, “Scalar perturbations in conformal rollingscenario with intermediate stage,” JCAP (2011) 010, arXiv:1102.1390 [hep-th] .[46] M. Libanov, S. Mironov, and V. Rubakov, “Non-Gaussianity of scalar perturbationsgenerated by conformal mechanisms,” Phys.Rev. D84 (2011) 083502, arXiv:1105.6230[astro-ph.CO] .[47] M. Osipov and V. Rubakov, “Galileon bounce after ekpyrotic contraction,” JCAP (2013) 031, arXiv:1303.1221 [hep-th] .[48] K. Hinterbichler and J. Khoury, “The Pseudo-Conformal Universe: Scale Invariance fromSpontaneous Breaking of Conformal Symmetry,” JCAP (2012) 023, arXiv:1106.1428[hep-th] .[49] K. Hinterbichler, A. Joyce, and J. Khoury, “Non-linear Realizations of Conformal Symmetryand Effective Field Theory for the Pseudo-Conformal Universe,” JCAP (2012) 043, arXiv:1202.6056 [hep-th] .[50] M. Koehn, J.-L. Lehners, and B. A. Ovrut, “A Cosmological Super-Bounce,” arXiv:1310.7577 [hep-th] .[51] J. De-Santiago, J. L. Cervantes-Cota, and D. Wands, “Cosmological phase space analysis ofthe F (X) - V (phi) scalar field and bouncing solutions,” Phys.Rev.
D87 (2013) 023502, arXiv:1204.3631 [gr-qc] .[52] A. Nayeri, R. H. Brandenberger, and C. Vafa, “Producing a scale-invariant spectrum ofperturbations in a Hagedorn phase of string cosmology,” Phys.Rev.Lett. (2006) 021302, arXiv:hep-th/0511140 [hep-th] . 2553] P. Creminelli, M. A. Luty, A. Nicolis, and L. Senatore, “Starting the Universe: StableViolation of the Null Energy Condition and Non-standard Cosmologies,” JHEP (2006)080, arXiv:hep-th/0606090 [hep-th] .[54] P. Creminelli, A. Nicolis, and E. Trincherini, “Galilean Genesis: An Alternative to inflation,”JCAP (2010) 021, arXiv:1007.0027 [hep-th] .[55] L. Perreault Levasseur, R. Brandenberger, and A.-C. Davis, “Defrosting in an EmergentGalileon Cosmology,” Phys.Rev. D84 (2011) 103512, arXiv:1105.5649 [astro-ph.CO] .[56] Z.-G. Liu, J. Zhang, and Y.-S. Piao, “A Galileon Design of Slow Expansion,” Phys.Rev.
D84 (2011) 063508, arXiv:1105.5713 [astro-ph.CO] .[57] Y. Wang and R. Brandenberger, “Scale-Invariant Fluctuations from Galilean Genesis,”JCAP (2012) 021, arXiv:1206.4309 [hep-th] .[58] Z.-G. Liu and Y.-S. Piao, “A Galileon Design of Slow Expansion: Emergent universe,”Phys.Lett.
B718 (2013) 734–739, arXiv:1207.2568 [gr-qc] .[59] P. Creminelli, K. Hinterbichler, J. Khoury, A. Nicolis, and E. Trincherini, “SubluminalGalilean Genesis,” JHEP (2013) 006, arXiv:1209.3768 [hep-th] .[60] K. Hinterbichler, A. Joyce, J. Khoury, and G. E. Miller, “DBI Realizations of thePseudo-Conformal Universe and Galilean Genesis Scenarios,” JCAP (2012) 030, arXiv:1209.5742 [hep-th] .[61] K. Hinterbichler, A. Joyce, J. Khoury, and G. E. Miller, “DBI Genesis: An ImprovedViolation of the Null Energy Condition,” Phys.Rev.Lett. (2013) 241303, arXiv:1212.3607 [hep-th] .[62] C. Deffayet, O. Pujolas, I. Sawicki, and A. Vikman, “Imperfect Dark Energy from KineticGravity Braiding,” JCAP (2010) 026, arXiv:1008.0048 [hep-th] .[63] S. Dubovsky, T. Gregoire, A. Nicolis, and R. Rattazzi, “Null energy condition andsuperluminal propagation,” JHEP (2006) 025, arXiv:hep-th/0512260 [hep-th] .[64] I. Sawicki and A. Vikman, “Hidden Negative Energies in Strongly Accelerated Universes,”Phys.Rev.
D87 (2013) no. 6, 067301, arXiv:1209.2961 [astro-ph.CO] .[65] V. Rubakov, “Consistent NEC-violation: towards creating a universe in the laboratory,” arXiv:1305.2614 [hep-th] .[66] C. de Rham and A. J. Tolley, “DBI and the Galileon reunited,” JCAP (2010) 015, arXiv:1003.5917 [hep-th] .[67] N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, “Ghost condensation and aconsistent infrared modification of gravity,” JHEP (2004) 074, arXiv:hep-th/0312099[hep-th] .[68] A. Nicolis, R. Rattazzi, and E. Trincherini, “Energy’s and amplitudes’ positivity,” JHEP (2010) 095, arXiv:0912.4258 [hep-th] .2669] S. Dubovsky and S. Sibiryakov, “Spontaneous breaking of Lorentz invariance, black holesand perpetuum mobile of the 2nd kind,” Phys.Lett.
B638 (2006) 509–514, arXiv:hep-th/0603158 [hep-th] .[70] P. Creminelli, M. Serone, and E. Trincherini, “Non-linear Representations of the ConformalGroup and Mapping of Galileons,” JHEP (2013) 040, arXiv:1306.2946 [hep-th] .[71] C. de Rham, M. Fasiello, and A. J. Tolley, “Galileon Duality,” arXiv:1308.2702 [hep-th] .[72] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, “Causality,analyticity and an IR obstruction to UV completion,” JHEP (2006) 014, arXiv:hep-th/0602178 [hep-th] .[73] D. A. Easson, I. Sawicki, and A. Vikman, “When Matter Matters,” JCAP (2013) 014, arXiv:1304.3903 [hep-th] .[74] V. Rubakov, “The Null Energy Condition and its violation,” arXiv:1401.4024 [hep-th] .[75] R. V. Buniy, S. D. Hsu, and B. M. Murray, “The Null energy condition and instability,”Phys.Rev.
D74 (2006) 063518, arXiv:hep-th/0606091 [hep-th] .[76] A. Nicolis, R. Rattazzi, and E. Trincherini, “The Galileon as a local modification of gravity,”Phys.Rev.
D79 (2009) 064036, arXiv:0811.2197 [hep-th] .[77] D. A. Easson, I. Sawicki, and A. Vikman, “G-Bounce,” JCAP (2011) 021, arXiv:1109.1047 [hep-th] .[78] P. Creminelli, A. Joyce, J. Khoury, and M. Simonovic, “Consistency Relations for theConformal Mechanism,” JCAP (2013) 020, arXiv:1212.3329 .[79] D. Lovelock, “The Einstein tensor and its generalizations,” J.Math.Phys. (1971) 498–501.[80] G. Goon, K. Hinterbichler, and M. Trodden, “Symmetries for Galileons and DBI scalars oncurved space,” JCAP (2011) 017, arXiv:1103.5745 [hep-th] .[81] M. Alishahiha, E. Silverstein, and D. Tong, “DBI in the sky,” Phys.Rev. D70 (2004) 123505, arXiv:hep-th/0404084 [hep-th] .[82] S. Mukohyama, “Ghost condensate and generalized second law,” JHEP (2009) 070, arXiv:0901.3595 [hep-th] .[83] C. Deffayet, G. Esposito-Farese, and A. Vikman, “Covariant Galileon,” Phys.Rev.
D79 (2009) 084003, arXiv:0901.1314 [hep-th] .[84] L. Berezhiani, Y.-Z. Chu, J. Khoury, and M. Trodden, “in progress,”.[85] M. Ostrogradski, “Memoires sur les equations differentielles relatives au probleme desisoperimetres,” Mem. Ac. St. Petersbourg
VI 4 (1850) 385.[86] R. P. Woodard, “Avoiding dark energy with 1/r modifications of gravity,” Lect.Notes Phys. (2007) 403–433, arXiv:astro-ph/0601672 [astro-ph]arXiv:astro-ph/0601672 [astro-ph]