The Null Energy Condition and its violation

TL;DR

This paper reviews scalar field theories with second-derivative Lagrangians that can violate the Null Energy Condition without obvious problems, exploring their applications in cosmology and universe creation experiments.

Contribution

It provides a concise overview of theories violating the Null Energy Condition and discusses their potential use in cosmology and laboratory universe creation.

Findings

Some scalar theories admit NEC violations without pathologies

Examples of these theories applied in cosmology

Potential for universe creation in laboratory settings

Abstract

We give a mini-review of scalar field theories with second-derivative Lagrangians, whose field equations are second order. Some of these theories admit solutions violating the Null Energy Condition and having no obvious pathologies. We give a few examples of using these theories in cosmological setting and also in the context of the creation of a universe in the laboratory.

Equations

$$T_{\mu \nu }{n}^{\mu }{n}^{\nu }>0,$$

$$d{s}^2=d{t}^2-a^2(t){\gamma }_{ij}d{x}^{i}d{x}^j,$$

$$T_{00}$$

$$T_{ij}$$

$$H^2$$

$$2\dot{H}+3H^2$$

$$\dot{H}=-4\pi G(\rho +p)+\frac{\kappa }{a^2}$$

$$\rho +p>0.$$

$$\frac{d\rho }{dt}=-3H(\rho +p).$$

$$L_{\chi }^{(2)}=\frac{1}{2}U{\dot{\chi }}^2-\frac{1}{2}V({\partial }_i\chi {)}^2-\frac{1}{2}W{\chi }^2,$$

$$U>0,V>0,W\ge 0.$$

$$U{\omega }^2=V{\mathbf{p}}^2+W,$$

$$T_{00}^{(2)}=\frac{1}{2}U{\dot{\chi }}^2+\frac{1}{2}V({\partial }_i\chi {)}^2+\frac{1}{2}W{\chi }^2$$

$$U>V>0.$$

$$U>0,V=0.$$

$$L_{\chi }^{(2)}=\frac{1}{2}U{\dot{\chi }}^2+\frac{1}{2{\Lambda }^2}\left[a{\ddot{\chi }}^2+b{\dot{\chi }}^2({\partial }_i\chi {)}^2+c({\partial }_i{\partial }_i\chi {)}^2\right],$$

$$U{\omega }^2=\frac{c}{{\Lambda }^2}{\mathbf{p}}^4,$$

$$U>0,V>0,W<0.$$

$$U>0,V<0,\text{mbox}orU<0,V>0.$$

$$U<0,V<0.$$

$$L_{\chi }^{(2)}=\frac{1}{2}{U}_{IJ}{\dot{\chi }}^I{\dot{\chi }}^J-\frac{1}{2}{V}_{IJ}{\partial }_i{\chi }^I{\partial }_i{\chi }^J-\frac{1}{2}{W}_{IJ}{\chi }^I{\chi }^J.$$

$$T_{00}^{(2)}=\frac{1}{2}{U}_{IJ}{\dot{\chi }}^I{\dot{\chi }}^J+\frac{1}{2}{V}_{IJ}{\partial }_i{\chi }^I{\partial }_i{\chi }^J+\frac{1}{2}{W}_{IJ}{\chi }^I{\chi }^J.$$

$$\text{mbox}Stablebackground:\text{mbox}positivedefinite{U}_{IJ},V_{IJ}.$$

$$L=F(X^{IJ},{\pi }^I),$$

$$X^{IJ}={\partial }_{\mu }{\pi }^I{\partial }^{\mu }{\pi }^J.$$

$$T_{\mu \nu }=2\frac{\partial F}{\partial X^{IJ}}{\partial }_{\mu }{\pi }^I{\partial }_{\nu }{\pi }^J-g_{\mu \nu }F.$$

$$T_{00}\equiv \rho$$

$$T_{11}$$

$$\rho +p=2\frac{\partial F}{\partial X^{IJ}}{X}^{IJ}=2\frac{\partial F}{\partial X^{IJ}}{\dot{\pi }}^I{\dot{\pi }}^J.$$

$$U_{IJ}$$

$$V_{IJ}$$

$$L=M^4(X^2-1{)}^2-V(\pi ),$$

Full text

1. 1 Introduction

2. 2 NEC-violation and instabilities

3. 2.1 Tachyons, gradient instabilities, ghosts

4. 2.2 Scalar theories with first-derivative Lagrangians

5. 3 Second-derivative Lagrangians

6. 4 Examples of NEC-violation

7. 4.1 Rolling background

8. 4.2 Genesis scenario

9. 4.3 Bouncing Universe

10. 4.4 Creating a universe in the Laboratory

11. 5 Conclusion

12. 6 Appendix A

13. 7 Appendix B

14. 8 Appendix C

1 Introduction

Among various energy conditions discussed in the context of General Relativity, the Null Energy Condition (NEC) plays a special role. This condition states that the matter energy-momentum tensor $T_{\mu\nu}$ obeys

[TABLE]

for any null (light-like) vector $n^{\mu}$, i.e., for any vector satisfying $g_{\mu\nu}n^{\mu}n^{\nu}=0$. The reason for the NEC being particularly interesting, is twofold. First, the NEC is quite robust; we illustrate this point in Section 2. In fact, until rather recently the common lore was that the NEC could not be violated in a healthy theory, with possible exception of scalar field non-minimally coupled to gravity [1]. The developments that refuted this viewpoint are the main emphasis of this mini-review.

Second, the NEC is a crucial assumption of the Penrose singularity theorem [2], valid in General Relativity. The theorem assumes that (i) the NEC holds; (ii) Cauchi hypersurface is non-compact. The theorem states that once there is a trapped surface in space, then there will be a singularity in future. A trapped surface is a closed surface on which outward-pointing light rays are actually converging (moving inwards). In a spherically symmetric situation, this means the following. Let $R$ be a coordinate that measures the area of a sphere, $S(R)=4\pi R^{2}$. Then a sphere is a trapped surface if $R$ decreases along any future null direction; all light rays emanating from this sphere in this sense move towards its center. See Apendix A for details. An example is a sphere inside the horizon of the Schwarzschild black hole, or in the case of contracting, spatially flat homogeneous isotropic Universe, a sphere of size greater than $|H|^{-1}$, where $H$ is the Hubble parameter. Thus, for matter obeying the NEC, there is always a singularity that gets formed inside a black hole horizon, and any contracting Universe ends up in a singularity, provided its spatial curvature is dynamically negligible (which is often the case). By time reversal, an expanding Universe has a singularity in the past. All this is true in classical General Relativity; things are different in other classical theories of gravity, and probably very different in quantum gravity.

In particular, the Penrose theorem almost forbids, within classical General Relativity, a bouncing Universe scenario, in which the Universe contracts at early times, contraction terminates at some moment of time and the Universe enters the expansion epoch which contitues until today. Let us see explicitly that the NEC is crucial for that ban. Consider a homogeneous isotropic Universe with Friedmann–Lemaître–Robertson–Walker metric

[TABLE]

where $\gamma_{ij}$ is time-independent metric of unit 3-sphere (one assigns a parameter $\kappa=+1$ to this case) or unit 3-hyperboloid ($\kappa=-1$) or Euclidean 3-dimensional space ($\kappa=0$). Matter governing the evolution of this Universe must also be homogeneous and isotropic, meaning that the only non-vanishing components of energy-momentum tensor are

[TABLE]

where $\rho$ and $p$ are energy density and effective pressure, respectively. $(00)$ and $(ij)$ components of the Einstein equations then give

[TABLE]

where $H\equiv\dot{a}/{a}$ is the Hubble parameter. A combination of these equations determines how it changes in time,

[TABLE]

Now, one chooses the null vector $n^{\mu}$ entering eq. (1) as $n^{\mu}=(1,a^{-1}\nu^{i})$, where $\gamma_{ij}\nu^{i}\nu^{j}=1$ and finds that the NEC is equivalent in the cosmological setting to

[TABLE]

Hence, if the second, spatial curvature term in the right hand side of eq. (4) is negaive ($\kappa<0$, open Universe), zero ($\kappa=0$, spatially flat Universe) or negligible, the Hubble parameter decreases in time. If it is negative (contraction), it remains negative. So, the bouncing Universe is almost impossible. A loophole is that the bounce is possible for closed Universe ($\kappa=+1$), provided that energy density and pressure grow slower than $a^{-2}$ as the Universe shrinks111According to eq. (5), for matter with equation of state $p=w\rho$ this requires $w<-1/3$.  [3]. Note that the Penrose theorem does not apply in the latter case, since the Cauchi hypersurface is compact in the closed Universe (3-sphere).

Applied to the present Universe (which is spatially flat to an excellent accuracy), the NEC implies that the Hubble parameter cannot grow today. Observational evidence for the growing Hubble parameter would mean that either dark energy violates the NEC or General Relativity is not valid at the present cosmological scales. This would of course be highly non-trivial.

Another face of the NEC shows up through the covariant energy-momentum conservation, $\nabla_{\mu}T^{\mu\nu}=0$. In the cosmological setting it reads

[TABLE]

Thus, the NEC implies that energy density always decreases in expanding Universe. Modulo the loophole mentioned above, the Penrose theorem states that the expansion started from a singularity – infinite energy density, infinite expansion rate.

One more consequence of the NEC is an obstruction for the creation of a universe in the laboratory. The question of whether one can in principle create a universe in the laboratory has been raised [4, 5] soon after the invention of inflationary theory [6]. Indeed, inflation – nearly exponential expansion of the Universe at high expansion rate – is capable of stretching, in a fraction of a second, a tiny region of space into a region of huge size, possibly exceeding the size of the presently observable Universe. So, it appears at first sight that it is not impossible to create artificially a region in our present Universe in which the physical conditions are similar to those at the onset of inflation, and then this region would automatically expand to very large size and become a universe like ours. In theories obeying the NEC and within classical General Relativity this is impossible [5, 7] because of the Penrose theorem. By definition, a universe “like ours” is a nearly homogeneous patch in space whose size exceeds the Hubble distance $H^{-1}$. Then the Hubble sphere is an anti-trapped surface, and hence there had to be a singularity in the past. Since we cannot create an appropriate singularity (and control the evolution through any singularity), we cannot create a universe “like ours”. Widely discussed ways out are to invoke tunneling [8, 9, 10, 11] or other quantum effects [12, 13, 14] and modify gravity [15, 16, 17], but it is certainly of interest to stay within General Relativity and invoke NEC-violation instead. There were several attempts in the latter direction [18], but many of them are problematic because of instabilities.

Finally, the NEC also forbids the existence, within General Relativity, of throats in space, both static [19, 20, 21] and time-dependent [22]. Such a throat could join asymptotically flat regions of space, forming a Lorenzian wormhole [19, 20, 21, 23, 24], Fig. 1. Alternatively, it could serve as a bridge between large but finite region of space and asymptotically flat region, forming a semi-closed world [12], Fig. 2. Again, it is of interest to construct healthy NEC-violating theories possessing wormhole solutions.

All this motivates one to search for healthy NEC-violating theories. For a field theorist, it is natural to start with scalar field theories. However, as we discuss in Section 2, solutions in theories of scalar fields minimally coupled to gravity and described by Lagrangians containing first derivatives only, either obey the NEC or have pathologies (modulo a loophole which we briefly discuss in Section 2). Because of that, one either turns to vector fields (and, indeed, there are examples of acceptable NEC-violating solutions in rather contrived theories involving vector, but not gauge, fields [25, 26]) or considers higher-derivative Lagrangians. It is commonly thought, however, that theories with Lagrangians containing second and higher derivatives are unacceptable (unless the higher derivative terms are treated as perturbations in the sense of effective low energy theory), since their field equations involve more than two derivatives, and hence these theories have pathological degrees of freedom. This is not the case, however: there exists a class of scalar field theories with second-derivative Lagrangians and yet second order field equations. These theories have been found in an unnoticed paper by G. W. Horndeski [27], rediscovered in rather different context by D. B. Fairlie, J. Govaerts and A. Morozov [28] (see Ref. [29] for a review of this approach) and relatively recently became popular in their various reincarnations, such as the Dvali–Gabadadze–Porrati [30] model in decouping limit [31, 32], Galileon theory [33] and its generalizations [34, 35, 36, 37, 38], k-mouflage [39], kinetic gravity brading [40, 41, 42], Fab-Four [43], etc. As we discuss in Section 3, there exist finite number of classes of second-derivative Lagrangians yielding second-derivative field equations [33, 44, 45, 46]. At least some of these Lagrangians admit NEC-violation [47, 48, 49, 50, 51, 52], with the NEC-violating solutions and their neighborhoods being perfectly healthy. In Section 4 we give a few examples of using these theories for constructing fairly non-trivial cosmological models and giving a proof-of-principle construction for creating a universe in the laboratory.

We conclude in Section 5 by pointing at some potentially problematic features of the NEC-violating second-derivative theories that have yet to be understood.

2 NEC-violation and instabilities

2.1 Tachyons, gradient instabilities, ghosts

In this paper we are mainly interested in weak gravity regime, which occurs when $M_{Pl}$ is the largest parameter in the problem. To the lowest order this corresponds to switching off dynamics of metric and considering other fields in Minkowski background. In many cases the relevant solutions to the fields equations are spatially homogeneous, and in this mini-review we stick to his case. We ask whether the NEC can be violated in this situation.

In a theory of one scalar field $\pi$, a spatially homogeneous classical solution $\pi_{c}(t)$ may or may not be pathological. The pathology, if any, shows up in the behavior of small perturbations about this background, $\pi=\pi_{c}+\chi$. Assuming that the linearized field equation for $\chi$ is second order in derivatives, the quadratic Lagrangian for $\chi$ always reads

[TABLE]

where $U$, $V$, $W$ depend on time. Let us consider high momentum regime, meaning that variations of $\chi$ in space and time occur at scales much shorter than the time scale charactersitic of the background $\pi_{c}(t)$. Then at a given time, the time-dependence of $U$, $V$ and $W$ can be neglected, and there are the following possibilities:

(1) Stable background:

[TABLE]

The dispersion relation is

[TABLE]

which is the dispersion relation for conventional excitations, while the energy density for perturbations

[TABLE]

is positive, as it should. For $V<U$, the $\chi$-waves travel at subluminal speed, for $V=U$ they travel at the speed of light, while for $V>U$ the $\chi$-waves are superluminal. While the superluminal propagation is probably less of a problem, it does signal that the theory cannot be UV-completed in a Lorentz-invariant way [53] (meaning that it cannot be a low energy theory of some Lorentz-invariant quantum theory valid at all scales), see, however, Ref. [54] which debates this point. So, one would like to avoid superluminality. The case $U=V$ is also potentially problematic, since there may or may not be backgrounds in a neighborhood of $\pi_{c}$, about which the perturbations are superluminal. So, the safe case is

[TABLE]

(1a) Special case:

[TABLE]

To understand how to treat this case, one thinks of the original scalar theory as an effective field theory with UV cutoff $\Lambda$. The Lagrangian of such a theory generically has corrections of higher order in derivatives, wich are suppressed by powers of $\Lambda^{-1}$ and hence are normally negligible. For $V=0$, however, these corrections cannot be neglected, since only these corrections give the terms in the Lagrangian for perturbations which involve spatial gradients [55]. The dominant higher derivative terms in the Lagrangian for perturbations involve second derivatives, so the Lagrangian reads222A possible lower derivative term $(\alpha(t)/\Lambda)\dot{\chi}(\partial_{i}\chi)^{2}$, upon integration by parts, reduces to $(\dot{\alpha}/2\Lambda)(\partial_{i}\chi)^{2}$; the pertinent transformation reads $(\alpha/\Lambda)\dot{\chi}(\partial_{i}\chi)^{2}\to-(\alpha/\Lambda)\partial_{i}\dot{\chi}\cdot\partial_{i}\chi=-(\alpha/2\Lambda)\partial_{0}(\partial_{i}\chi)^{2}\to(\dot{\alpha}/2\Lambda)(\partial_{i}\chi)^{2}$. It is subdominant at ${\bf p}^{2}\gg\dot{\alpha}{\Lambda}$, but becomes relevant at lower momenta. It is healthy for $\dot{\alpha}<0$.

[TABLE]

where we set $W=0$ for simplicity, as in the ghost condensate theory [55]. The dispersion relation, modulo corrections even stronger suppressed by $\Lambda^{-1}$, is now

[TABLE]

which is healthy for $c>0$. Other roots of the dispersion equation obey $|\omega|\gg\Lambda$, so they cannot be trusted in the low energy effective theory.

(1b) Tachyonic instability:

[TABLE]

Formally, the dispersion relation (7) yields imaginary $\omega$ for sufficiently low momenta, $V{\bf p}^{2}<|W|$, so that there are growing perturbations, $\chi\propto\exp\left(\int|\omega|dt\right)$ with $|\omega|\leq|W|^{1/2}$. This is indeed a problem, if the time scale $|W|^{-1/2}$ is much shorter than the time scale characteristic of the background $\pi_{c}(t)$. In the opposite case one cannot use the approximation of slowly varying $U(t)$, $V(t)$ and $W(t)$ and hence cannot conclude that the background $\pi_{c}$ is unstable. Instead, the background is stable at short time scales, and to see what is going on at long time scales one has to perform full stability analysis. We note in passing that tachyonic instabilities are inherent in some NEC-violating models of dark energy [25, 26], and they may have interesting observational consequences [56, 57].

(2) Gradient instability:

[TABLE]

According to eq. (7), “frequencies” $\omega({\bf p})$ are imaginary at high momenta, and there are perturbations that grow arbitrarily fast. This means that the background $\pi_{c}$ is unstable, and thus not healthy. Considering the original scalar theory as an effectve low energy theory theory valid below a certain UV scale $\Lambda$ does not help: for consistency, the rate of variation of the background $\pi_{c}(t)$ must be well below $\Lambda$, while the rates of development of the instabilities extend up to $\Lambda$; the background is ruined at short time scale.

(3) Ghost instability

[TABLE]

In classical field theory, the background is stable against high momentum perturbations: eq. (7) shows that the frequencies are real at high momenta. Yet the background is quantum-mechanically unstable. Indeed, the energy (8) is negative at high momenta, and upon quantization the $\chi$-particles have negative energies; they are ghosts. The energy conservation does not forbid pair creation from vacuum of ghosts together with other, normal particles (say, via graviton exchange, since gravitons definitely interact with $\chi$-quanta); vacuum is quantum-mechanically unstable. Energies and momenta of created particles can take values up to the UV scale $\Lambda$ below which one can trust the theory, so the available phase space is generically large, and the time scale of instability is short. Unless $\Lambda$ is low enough, this instability is unacceptable. So, backgrounds with ghosts are generally considered as pathological. We note in passing that in Lorentz-invariant theory and for Lorentz-invariant background $\pi_{c}=\mbox{const}$, the ghost instability is truly catastrophic: if particles can be created from vacuum with some energies and momenta, then the same, but Lorentz-boosted process is also allowed; the available phase space is proportional to the volume of the Lorentz group, i.e., it is infinite; the time scale of instability is infinitesimally short. Put it differently, ghosts in the present Universe are allowed only if Lorentz-invariance is violated in the ghost sector in such a way that energies of ghost particles cannot exceed $3$ MeV [58].

The above discussion is straightforwardly generalized to a theory with several scalar fields $\pi^{I}$, $I=1,\dots,N$. The Lagrangian for perturbations $\chi^{I}$ is now

[TABLE]

and the energy density is

[TABLE]

Barring the case of degenerate matrix $V_{IJ}$, similar to (1a) above, the matrix $V_{IJ}$ can be diagonalized by field redefinition. If it has negaive eigenvalue(s), the energy is unbounded from below [59]: one can construct an initial configuration with $\dot{\chi}^{I}=0$ with arbitrarily high momentum and ${\bf p}^{2}V_{IJ}\chi^{I}\chi^{J}<0$. This is a pathological situation: there are either ghosts or gradient instabilities, or both. For positive definite diagonal $V_{IJ}$ one can rescale $\chi^{I}$ to cast $V_{IJ}$ into unit matrix, $V_{IJ}=\delta_{IJ}$. One can then diagonalize $U_{IJ}$ by orthogonal transformation, so the derivative terms in the Lagrangian become $\sum_{I}\left[\lambda_{I}(\dot{\chi}^{I})^{2}-(\partial_{i}\chi^{I})^{2}\right]$. If $U_{IJ}$ has negative eigenvalues $\lambda_{I}$, there are gradient instabilities. So, the requirement of the absence of gradient instabilities and ghosts gives the necessary condition

[TABLE]

Whether or not there are tachyons at sufficiently low momenta depends now on positive definiteness of $W_{IJ}$.

2.2 Scalar theories with first-derivative Lagrangians

As the first attempt to construct NEC-violating theory, one considers the Lagrangian involving first derivatives only,

[TABLE]

where

[TABLE]

One assumes minimal coupling to gravity, then the energy-momentum tensor for this theory reads

[TABLE]

Therefore, for homogeneous background

[TABLE]

and

[TABLE]

We see that NEC-violation requires that the matrix $\partial F/\partial X_{c}^{IJ}$, evaluated for the background $\pi_{c}^{I}$, be non-positive definite. On the other hand, one expands the Lagrangian (11) to the second order in perturbations $\pi^{I}=\pi_{c}^{I}+\chi^{I}$ and obtains the Lagrangian for perturbations in the form (9) with

[TABLE]

Thus, the stability of the background – positive definiteness of $V_{IJ}$, eq. (10) – is inconsistent with NEC-violation [59].

A loophole here is related to the case (1a) above [60]. To this end, consider ghost condensate theory with small potential added [60, 61],

[TABLE]

where $\pi$ is the ghost condensate field (of dimension $(\mbox{mass})^{-1}$), $X=\partial_{\mu}\pi\partial^{\mu}\pi$ and $M$ is the energy scale. In the absence of the potential, there is a solution $\pi_{c}=t$ for which $F\equiv M^{4}(X^{2}-1)^{2}=0$ and $\partial F/\partial X=0$. This is on the borderline of NEC-violation. Higher derivative term of appropriate sign renders this background stable. Now, upon adding small potential $V(\pi)$ with positive slope, one makes $(\dot{\pi}_{c}-1)$ slightly negative. According to eqs. (12) and (13), this leads to NEC-violation, and at the same time to the gradient instability. However, with the higher derivative terms present, the latter instability occurs at low momenta ${\bf p}$ only, and can be made harmless [61] by careful choice of parameters and of the form of the higher derivative corrections. This construction was used in Ref. [61], in particular, to design a viable cosmological scenario similar to what is now called Genesis. We discuss a less contrived Genesis model in Section 4. Also, ghost condensate idea was used to construct consistent bouncing Universe models [62, 63], which start from the ekpyrotic contraction stage [64, 65]. Again, consistency of the bounce requires careful choice of parameters in these models. We consider a simpler version of this scenario in Section 4.

3 Second-derivative Lagrangians

The main emphasis of this mini-review is on scalar field theories with Lagrangians involving second derivatives, whose equations of motion do not contain third and fourth derivatives nevertheless. Although nomenclature has not yet been settled, we call them (generalized) Galileons. We concentrate on theories of one scalar field $\pi$ in Minkowski space and write the Euler–Lagrange equation for a theory with the Lagrangian $L(\pi,\partial_{\mu}\pi,\partial_{\mu}\partial_{\nu}\pi)$:

[TABLE]

where

[TABLE]

Because of the last term in eq. (14), the field equation is generically of fourth order in derivatives. However, there are exceptions, which are precisely Galileons. The simplest exceptional second derivative Lagrangian is

[TABLE]

Off hand, the corresponding field equation is third order, but in fact it is not. Indeed, the second term in eq. (14) gives rise to the folowing third order contribution

[TABLE]

while the third term in eq. (14) reads

[TABLE]

where omitted terms do not contain third derivatives. Hence, third order terms cancel out, and the field equation is second order.

It is instructive to make the following observation. There appear to be two terms of the general form (15) with different Lorentz structure:

[TABLE]

where $\Box=\partial_{\lambda}\partial^{\lambda}$ and, as before, $X=\partial_{\lambda}\pi\partial^{\lambda}\pi$. However, the second structure can be reduced to the first one by integrating by parts (which we denote by arrow):

[TABLE]

where the function $Q(\pi,X)$ is such that $H=\partial Q/\partial X$. So, the only remaining term in the Lagrangian is

[TABLE]

Note this term cannot be rediced by integration by parts to any Lagrangian involving first derivatives only.

Let us consider a more complicated example of the Lagrangian quadratic in the second derivatives. There are five possible Lorentz structures:

[TABLE]

where $F_{a}=F_{a}(\pi,X)$, $a=1,\dots,5$. The resulting field equation has the following fourth order terms

[TABLE]

We see that the fourth order terms cancel out iff $F_{1}=0$, $F_{2}=-F_{3}$, $F_{4}=-F_{5}$, so that the Largagian has the following form

[TABLE]

where square brackets denote anti-symmetrization (our definition is $A_{[\mu\nu]}=A_{\mu\nu}-A_{\nu\mu}$ without numerical prefactor). We now understand the reason for the cancellation of the fourth order terms in the field equation: it happens because, e.g., $\partial_{\mu}\partial_{\left[\nu\right.}\partial_{\left.\lambda\right]}\partial^{\mu}\pi=0$. Again, the first term in (21) can be cast into the form of the second term:

[TABLE]

So, there again remains one term

[TABLE]

Now it is straightforward to check that the third order terms in the field equation also cancel out: the terms with $\left(\partial K_{2}/\partial\pi_{\lambda}\right)\partial_{\lambda}\left(\partial^{\nu}\partial_{\nu}\pi\partial_{\mu}\partial^{\mu}\pi-\partial^{\nu}\partial_{\mu}\pi\partial_{\nu}\partial^{\mu}\pi\right)$ cancel out automatically in the same way as in eqs. (16), (17), while the remaining terms like

[TABLE]

also vanish.

The story repeats itself in cubic and higher orders in the second derivatives. The only exceptional $n$-th order term in $D$ dimensions is [33, 44]

[TABLE]

The fact that the corresponding field equation is second order is checked trivially; the proof that no other terms exist is not so simple [44] . Note that $L_{(n)}$ can be written as

[TABLE]

Indeed, any antisymmetric tensor $A_{\mu_{1}\dots\mu_{n}}$ can be written as

[TABLE]

where

[TABLE]

is the dual tensor. The expression in the right hand side of eq. (22) is antisymmetric in both upper and lower indices. Applying the transformation (24), (25) to upper indices and to lower indices separately, one arrives at the form (23).

Note that there are $(D+1)$ allowed classes of Lagrangians, if one counts the class without second derivatives

[TABLE]

In particular, there are five classes in four dimensions. A general Galileon Lagrangian is a sum of all these terms.

This completes the discussion of the exceptional theories of one scalar field, Galileon, in Minkowski space. Theories with multiple scalar fields are considered in Refs. [45, 46] (see also Refs. [66, 67, 68]). The minimal generalization of $L_{(1)}$ to curved space-time is simple:

[TABLE]

where $X=g^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi$; it is straightforward to check that the resulting field equation is still second order. The energy-momentum tensor, and hence the Einstein equations, are second oreder in derivatives as well. The generalizations of $L_{(2)}$ and higher order Lagrangians are, on the other hand, non-trivial [27, 34, 44]. Finally, we note that some Galileon Lagrangians have an interesting interpretation as describing a three-brane evolving in five-dimensional space-time [35, 36].

4 Examples of NEC-violation

In this Section we consider an example of simple NEC-violating solution and its use for constructing rather non-trivial cosmological scenarios. We also discuss the possibility of creatng a universe in the laboratory by employing the Galileon models of Section 3. Our set of illustrations is of course personal and by no means complete.

4.1 Rolling background

The analysis is particularly simple in models exhibiting scale invariance,

[TABLE]

It is sufficient for our purposes to consider the Lagrangian involving only the terms $L_{(0)}$ and $L_{(1)}$, eqs. (26) and (18). Theories of this general form have been studied in detail in Refs. [40, 41, 42] and named kinetic gravity braiding. We write in the scale-invariant case and in Minkowski space

[TABLE]

where

[TABLE]

and $F$ and $K$ are yet unspecified functions. Assuming that $K$ is analytic near the origin, we set

[TABLE]

Indeed, upon integrating by parts, a constant part of $K$ can be absorbed into the $F$-term in eq. (28). We will need the expression for the energy-momentum tensor. To this end, we consider minimal coupling to the metric, i.e., set $Y={\rm e}^{-2\pi}g^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi$ and $\Box\pi=\nabla^{\mu}\nabla_{\mu}\pi$ in curved space-time. To calculate the energy-momentum tensor, we note that in curved space-time, the $K$-term in $\sqrt{-g}L_{\pi}$ can be written, upon integrating by parts, as $\sqrt{-g}g^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\left(K{\rm e}^{2\pi}\right)$. Then the variation with respect to $g^{\mu\nu}$ is straightforward, and we get

[TABLE]

This expression is valid in curved space-time as well.

In what follows we consider homogeneous backgrounds in Minkowski space, $\pi=\pi(t)$. For a homogeneous field, the field equation reads

[TABLE]

while the energy density and pressure are

[TABLE]

where

[TABLE]

It is straightforward to see that for $\dot{\pi}\neq 0$, eq. (31) is equivalent to energy conservation, $\dot{\rho}=0$.

It is instructive to calculate the quadratic Lagrangian for perturbations about the homogeneous background. It has the form (6) with

[TABLE]

We will not need the general expression for $W$. Note that $U$ is proportional to the derivative with respect to $Y$ of the same function $Z$ that determines the energy density, eq. (32a).

With $F(0)=0$, the theory admits a constant solution $\pi_{c}=\mbox{const}$, $Y=0$ and $T_{\mu\nu}=0$. In the absence of other forms of energy, this solution corresponds to Minkowski space. Equations (33a), (33b) show that the Minkowski background is stable for $F^{\prime}(0)>0$, and that perturbations travel with the speed of light (recall that we set $K(0)=0$). This is easy to understand: it follows from eq. (28) that perturbations about constant $\pi_{c}$ are governed by the first term there, and $L^{(2)}={\rm e}^{2\pi_{c}}F^{\prime}(0)(\partial\chi)^{2}$, which is the Lagrangian for a massless scalar field. In the neighbourhood of the Minkowski background, i.e., for small $\partial\pi_{c}$, perturbations are not superluminal [51] provided that $K^{\prime}(0)=0$, $F^{\prime\prime}(0)>0$.

In a wide range of the functions $F$ and $K$, eq. (31) admits also a rolling solution,

[TABLE]

where $t_{*}$ is an arbitrary constant. For this solution $Y=Y_{*}=\mbox{const}$, and $Y_{*}$ is determined from equation

[TABLE]

where $F$, $F^{\prime}$, etc., are evaluated at $Y=Y_{*}$. For this solution one has $T_{00}=\rho=0$ and

[TABLE]

Thus, the rolling background violates the NEC, provided that

[TABLE]

The quadratic Lagrangian for perturbations (6) reduces in this background to

[TABLE]

where

[TABLE]

are time-independent coefficients. As a cross check, one can derive from the latter Lagrangian the equation for homogeneous perturbation $\chi(t)$ about the rolling background and see that $\chi=\partial_{t}\pi_{c}=(t_{*}-t)^{-1}$ obeys this equation, as it should. Indeed, making use of eq. (35) one finds that the coefficients of $\dot{\chi}^{2}$ and $\chi^{2}$ in eq. (38) are related in a simple way,

[TABLE]

Hence, homogeneous perturbation obeys a universal equation

[TABLE]

whose solutions are $\chi=(t_{*}-t)^{-1}$ and $\chi=(t_{*}-t)^{4}$. This shows that the rolling background is an attractor and that it is stable against low momentum perturbations: the growing perturbation $\chi=(t_{*}-t)^{-1}\cdot\chi_{0}({\bf x})$ with slowly varying $\chi_{0}({\bf x})$ can be absorbed into slightly inhomogeneous time shift.

Let us consider the stability of the rolling background and subluminality of the perturbations about it. The spatial gradient term in (38) has correct (negative) sign provided that

[TABLE]

The speed of perturbations about the rolling background is smaller than the speed of light, if the coefficient of $\dot{\chi}^{2}$ is greater than that of $-(\partial_{i}\chi)^{2}$, i.e.,

[TABLE]

We require that this inequality holds in strong sense, then the perturbations about the rolling solution are strictly subluminal, and hence the perturbations about backgrounds neighbouring the rolling solution are subluminal as well. When both inequalities (39) and (40) are satisfied, there are no ghosts either. The conditions (37), (39) and (40) together with eq. (35) can be satisfied at $Y=Y_{*}$ by a judicious choice of the functions $F$ and $K$ in the neighbourhood of this point, so that the NEC-violation is stable and subluminal. This can be seen as follows. Equation (35) can be used to express $F(Y_{*})$ in terms of $F^{\prime}(Y_{*})$, $K(Y_{*})$ and $K^{\prime}(Y_{*})$, namely, $F=2Y_{*}F^{\prime}-2Y_{*}K+2Y_{*}^{2}K^{\prime}$. Then the inequalities (37), (39) are satisfied, provided that $2K-4Y_{*}K^{\prime}<F^{\prime}<2K-Y_{*}K^{\prime}$, which is possible for positive $K^{\prime}$. The condition (40) can be satisfied by an appropriate choice of $F^{\prime\prime}$ and $K^{\prime\prime}$.

To make contact with existsing studies, we note that a particular Lagrangian of the type (28) considered in Ref. [48] is

[TABLE]

which corresponds to

[TABLE]

Here the parameters $f$ and $\Lambda$ have dimension of mass, the parameter $\alpha$ is dimensionless. The solution to eq. (35) is

[TABLE]

One requires that the energy scale $\sqrt{Y_{*}}$ associated with this solution is lower than $\Lambda$, which is interpreted as the UV cutoff scale. This gives

[TABLE]

From eq. (37) one finds that the background $Y=Y_{*}$ violates the NEC iff

[TABLE]

while the stability and subluminality conditions, eqs. (39) and (40), give

[TABLE]

All these conditions are satisfied for [48]

[TABLE]

Note that the case $\alpha=0$ corresponds to luminal propagation of perturbations about the background $Y=Y_{*}$. In fact, in this case the theory (41) is invariant under conformal symmetry [33, 47]. However, the case $\alpha=0$ is problematic, since there are backgrounds in the neighborhood of $Y=Y_{*}$ about which the propagation of perturbations is superluminal [48]. Note also that the Lagrangian (41) does not admit stable Minkowski background, since $F^{\prime}(0)<0$. Conformally invariant theory with stable Minkowski background and subluminal propagation about the solution $Y=Y_{*}$ and in its neighborhood was constructed in Ref. [50] building upon Ref. [35], and goes under the name DBI conformal Galileon theory.

To end up this Section, let us consider [51] the structure of the configuration space $(\pi,\dot{\pi})$ of spatially homogeneous Galileons in arbitrary Galileon theory with scale invariance (27). The Lagrangian may contain all terms discussed in Section 3. We pointed out above that for $\dot{\pi}\neq 0$, the field equation is equivalent to energy conservation, $\dot{\rho}=0$. This is not an accident. The Noether theorem states that the Noether energy-momentum tensor (which coincides with the metric energy-momentum tensor for the scalar field minimally coupled to gravity) obeys

[TABLE]

where $(\mbox{E.O.M.})$ stands for the equation of motion. Therefore, the equation of motion for spatially homogeneous $\pi=\pi(t)$ is

[TABLE]

Since the field equation is second order, $\rho=\rho(\pi,\dot{\pi})$ does not contain $\ddot{\pi}$ and higher derivatives, and by scale invariance it has the form

[TABLE]

where $Y=\dot{\pi}^{2}\,\exp(-2\pi)$, cf. eq. (29), and $Z$ is a model-dependent function. Now we can understand in more general terms that the rolling background with $Z=0$ and $\dot{\pi}>0$ is an attractor in the class of homogeneous solutions. To this end, we use the conservation of energy $\dot{\rho}=0$ and write for any homogeneous solution

[TABLE]

As $\pi$ increases, $|Z|$ decreases, so the solution tends to a configuration with $Z\to 0$. The configuration space of homogeneous Galileons with $\dot{\pi}>0$ is thus divided into basins of attraction of solutions with $Z=0$.

We also pointed out above that the coefficient $U$ entering the quadratic action for perturbations is proportional to $Z^{\prime}$. This is not an accident either. To see this, let us again use eq. (43) valid for any homogeneous Galileon. It follows from this equation that the equation of motion for homogeneous perturbation about the background $\pi_{c}(t)$ reads

[TABLE]

where omitted terms do not contain $\ddot{\chi}$. Hence, the Lagrangian for perturbations has the form

[TABLE]

where omitted terms do not contain $\dot{\pi}$. We conclude that $\rho={\rm e}^{4\pi_{c}}\,Z(Y_{c})$, $U={\rm e}^{2\pi_{c}}Z^{\prime}(Y_{c})$ for any point in the configuration space of homogeneous Galileon $(\pi_{c},\dot{\pi}_{c})$ in any scale-invariant Galileon theory.

Recall finally that a point in the configuration space $(\pi_{c},\dot{\pi}_{c})$, at which $U<0$, is unstable: there is either ghost or gradient instability among perturbations about this point. The above results therefore mean that any path in the space of homogeneous configurations $(\pi,\dot{\pi})$ that connects two zero energy attractor solutions, $Z=0$, passes through an unstable region: indeed, $Z^{\prime}$ is negative somewhere at this path. This property creates difficulties in using scale-invariant Galileons, as we discuss later on. Here we note that it implies that there is no evolution without pathologies that connects the Minkowski and rolling backgrounds, even if this evolution is driven by a source (provided that this source does not couple to $\dot{\pi}$).

The above analysis heavily uses scale invariance. Once one gives up scale invariance, this analysis and its conclusions are no longer valid. In particular, evolution from nearly Minkowski regime to rolling regime can occur without pathologies [52].

4.2 Genesis scenario

As the first example of utilizing the solution discussed in Section 4.1, let us consider Galilean Genesis [47] – a cosmological scenario alternative to inflation(see also Refs. [48, 69, 70, 49, 50, 71]). One assumes that at early times $t\to-\infty$, the space-time is Minkowskian, energy and pressure vanish, the Universe is empty. At that time the only relevant form of matter is the Galileon field $\pi$ described by the Lagrangian (28) (other Galileon Lagrangians are considered in Refs. [69, 50, 72] with fairly similar results). Once the conditions (37), (39) and (40) are satisfied, the solution $Y=Y_{*}$ is stable and violates the NEC. At the initial stage of evolution, i.e., at large enough $(t_{*}-t)$, energy density and pressure are small, and one can make use of the perturbation theory in $G\equiv M_{Pl}^{-2}$. Equation (4) with $\kappa=0$ determines the Hubble parameter, and to the lowest non-trivial order in $M_{Pl}^{-2}$ one makes use of the Minkowski expressions for energy density, $\rho=0$, and pressure, eq. (36),

[TABLE]

where $P=(2Y_{*}K-F)Y_{*}^{-2}$. One finds

[TABLE]

Equation (3a) is then used to find the energy density to the first order in $M_{Pl}^{-2}$:

[TABLE]

We see that as the field $\pi_{c}$ evolves, energy density builds up, and the cosmological expansion gets accelerated. The weak gravity approximation (expansion in $M_{Pl}^{-2}$) is valid when $\rho\ll p$, i.e.,

[TABLE]

The parameter $P$ may be large: in the example with the Lagrangian (41) one has $P\sim f^{3}/\Lambda^{3}\gg 1$ in view of eq. (42). Still, if $P$ is not exceedingly large, the weak gravity regime holds almost to the Planck scale.

As a cross check, let us consider the field equation for homogeneous $\pi$ in expanding spatially flat Universe. It reads

[TABLE]

We see that gravitational corrections here are small provided that $H\ll\dot{\pi}$, which again gives the condition (46). Discussing weak gravity regime is sufficient for our purposes, but of course one can follow the evolution after the end of this regime, with gravity effects fully accounted for. This is done in Ref. [47] in the model (41) with $\alpha=0$.

So far we have seen that the theory admits a cosmological scenario in which the Universe starts empty and Minkowskian and evolves into the stage of rapid expansion and high energy density. This evolution is precisely the Genesis epoch. There are two other ingredients in the Genesis scenario. First, at some late stage the Galileon energy density should be converted into heat, and the standard hot epoch should begin. A possible mechanism of “defrosting” is suggested in Ref. [73]. At the end of “defrosting” stage, whatever it is, the Galileon should settle to its Minkowski value, $\dot{\pi}=0$. In the scale-invariant Galileon theory this is problematic because of our observations in the end of Section 4.1. The violation of scale invariance at “defrosting” can probably cure this problem.

The second ingredient is a mechanism of the generation of density perturbations, responsible in the end for CMB anisotropies and structure formation. These perturbations are Gaussian (or nearly Gaussian) random field with nearly flat power spectrum. Perturbations in the Galileon field itself cannot do the job [47]. A simple extension of the Galileon theory can, however, work quite well [47] (see Refs. [69, 70] for alternative proposals). One insists on scale invariance at the Genesis epoch and adds a new field $\theta$ which trivially transforms under scale transformations, $\theta(x)\to\theta(\lambda x)$. By scale invariance, the kinetic term in its Lagrangian is

[TABLE]

If other interactions of the new field are negligible at the Genesis epoch, the Lagrangian in the rolling background (34) is

[TABLE]

This coincides with the Lagrangian of a scalar field minimally coupled to gravity, evolving at inflationary epoch with the Hubble parameter $\sqrt{Y_{*}}$, if one identifies $t$ with conformal time at inflation. Thus, one borrows the well-known result of the inflationary theory: vacuum fluctuations of the field $\theta$ develop into Gaussian random field with the power spectrum

[TABLE]

The field perturbations $\delta\theta$, which are entropy fluctuations at the Genesis epoch, are assumed to be reprocessed into adiabatic perturbations sometime after the Genesis epoch by, say, curvaton [74] or modulated decay [75] mechanism. The adiabatic perturbations $\zeta$ inherit the properties of perturbations $\delta\theta$ (modulo non-Gaussianities that may be produced in the process of conversion of entropy to adiabatic perturbtion); in particular, their power spectrum is ${\cal P}_{\zeta}=\mbox{const}\cdot{\cal P}_{\theta}$. The spectrum (48) is flat; small tilt, required by observations [76, 77], can emerge due to weak explicit breaking of scale invariance (cf. Ref. [78]).

To conclude this Section, we note that the Genesis scenario, especially its version with conformal Galileon, is an example of what is now called (pseudo-)conformal cosmology [79, 47, 80, 49]. In general terms, this class of scenarios assumes that the Universe is initially effectively Minkowskian, and matter is in conformally invariant state. Then conformal invariance is spontaneously broken by rolling background similar to (34). The mechanism of the generation of density perturbations is similar to one just discussed. Conformal scenario makes a number of model-independent predictions which potentially distinguish it from inflation. These include non-Gaussianities and statistical anisotropy of scalar perturbations [81, 82]. Another property is the absence of tensor perturbations.

4.3 Bouncing Universe

Galileon theories can also be used to construct models of bouncing Universe [47, 83, 84, 85, 86, 87, 88, 89]. Before we discuss a concrete model of this sort, let us make the following comment. Contracting Universe can easily become strongly inhomogeneous and anisotropic because of the Belinsky–Lifshits–Khalatnikov phenomenon [90]. This makes a problem for consistency of the entire bouncing scenario. A way to cure this problem is to assume that the dominant matter at the contracting stage has super-stiff equation of state, $p>\rho$ [91]. This is what one generically calls ekpyrotic Universe [64, 65]. We discuss this point in Appendix B. Note that for matter with the equation of state $p=w\rho$, $w=\mbox{const}$, eq. (5) gives $\rho\propto a^{-3(1+w)}$, and then one finds from eq. (3a) with $\kappa=0$ that the scale factor evolves as

[TABLE]

where

[TABLE]

Super-stiff equation of state, $w>1$, thus corresponds to

[TABLE]

An example of super-stiff matter is a scalar field with the negative exponential potential,

[TABLE]

where $V_{0}$ and $M$ are positive parameters. The equation for the homogeneous field $\phi(t)$ and the Friedmann equation(3a) have the following solution,

[TABLE]

where

[TABLE]

This is an attractor in the case of collapse. According to (49) and (52), the effective equation of state is indeed super-stiff, $w\gg 1$, for $M\ll M_{Pl}$. Note that the energy density is positive and increases as the Universe collapses,

[TABLE]

This leaves open the possibility that the potential $V(\phi)$ becomes positive at large $\phi$, and the field moves out of the negative potential at some late epoch.

It is worth noting that for $M\ll M_{Pl}$ this solution is always in the weak gravity regime similar to that studied in Section 4.2. In the weak gravity limit one neglects gravity in the field equation for $\phi$ and obtains the solution in Minkowski space

[TABLE]

The energy density vanishes in this limit, while pressure is

[TABLE]

The weak gravity approximation is valid at all times for $M\ll M_{Pl}$.

While one can construct models with just one scalar field, in which ekpyrotic contraction ends up in a bounce [85, 86, 88, 89], it is a lot simpler [87] to extend the model (50) by adding a Galileon field with the Lagrangian (28), so that the total matter Lagrangian is

[TABLE]

In the weak gravity limit, the fields $\phi$ and $\pi$ do not interact with each other, the Galileon rolls as in eq. (34), while $\phi(t)$ is given by (53). Energy density is zero, and pressure is the sum of (45) and (54):

[TABLE]

The Hubble parameter is found from eq. (4):

[TABLE]

At early times, the field $\phi$ dominates and the Universe contracts ($H<0$), later on the Galileon takes over, at least for $t_{*}<0$, the contraction terminates ($H=0$, bounce), the expansion epoch begins and proceeds like in the Genesis scenario ($H>0$). It is straightforward to see that the bounce indeed occurs in the weak gravity regime, $H\ll\dot{\pi}$, provided that the following mild inequality holds, $|t_{*}|\gg P^{1/2}M^{2}/M_{Pl}^{3}$, $t_{*}<0$ (the case $t_{*}>0$ is considered in Ref. [87] with the result that the bounce is always there, but it occurs not necessarily in the weak gravity regime).

To make this toy model more realistic, one modifies the potential $V(\phi)$ at large $\phi$ and adds the potential to the Galileon to ensure that the cosmological constant vanishes at late times. Depending on parameters, the system may or may not enter the late time inflationary regime [87]. The ingredients discussed in the end of Section 4.2 have to be present in this model as well.

4.4 Creating a universe in the Laboratory

Our last example is an attempt to design a model for the creation of a universe in the laboratory [51]. The idea is to construct initial condition in a Galileon-type theory such that inside some large sphere the field $\pi$ is nearly homogeneous and behaves like at the initial stage of Genesis, whereas outside this sphere this field tends to a constant and space-time is asymptotically Minkowskian. For this initial data, the energy density and pressure are initially small everywhere and the entire space-time is nearly Minkowskian, so that the required field configuration can in principle be prepared in the laboratory. As the field $\pi(t,{\bf x})$ evolves from this initial state according to its equation of motion, the energy density inside the large sphere increases, space undergoes accelerated expansion there, and the region inside the sphere eventually becomes a man-made universe. Outside this sphere the energy density remains small and asymptotes to zero at large distances; the space-time is always asymptotically Minkowskian.

It is tempting to implement this idea in a simple way, by considering the initial field $\pi(t,{\bf x})$ which slowly varies in space and interpolates between the rolling solution (34) inside the large sphere and Minkowski vacuum $\partial\pi=0$ at spatial infinity. By slow variation in space we mean that the spatial derivatives of $\pi$ are negligible compared to temporal ones, so that at each point in space $\pi$ evolves in the same way as in the homogeneous case.

An advantage of this quasi-homogeneous approach is its simplicity; a disadvantage is that it actually does not work in the class of scale-invariant models of Section 4.1. The obstruction comes from the property discussed in the end of Section 4.1: if the evolution of $\pi$ is effectively homogeneous everywhere, then the analysis of Section 4.1 applies, and since $Z(Y)$ vanishes both inside the large sphere (Genesis region) and far away from it (Minkowski region), there is a region in between where $Z^{\prime}<0$ and the system is unstable.

One way to get around this obstruction would be to insist on slow spatial variation of the initial field configuration but give up the prescription that the field inside the large sphere is in the Genesis regime (34). Instead, one would consider the field with non-zero energy density inside the sphere, so that there exists a smooth and stable configuration that interpolates, as $r$ increases, between this field and the asymptotic Minkowski vacuum. This can hardly lead to the creation of a universe, however, since, as we discussed in Section 4.1, the Minkowski point $Y=0$ is an attractor, and the field in the interior of the sphere will relax to it.

Other possibilities are to consider field configurations with non-negligible spatial gradients or give up scale-invariance of the action (the latter possibility has been successfully explored in Ref. [52] in the cosmological context). In either case the above no-go agrument would be irrelevant, but the analysis would be more complicated. It is simpler to follow another route, and complicate the model instead.

To this end, one allows the functions $F$ and $K$ to depend explicitly on spatial coordinates. This can be the case if there is another field, call it $\varphi$, which determines the couplings entering these functions, and this field acts as time-independent background, $\varphi=\varphi({\bf x})$. In this case one can consider a field configuration $\pi(t,{\bf x})$ which at any point in space is approximately given by the rolling solution (34), but with $Y_{*}$ depending on ${\bf x}$. One prepares the background $\varphi({\bf x})$ in such a way that $Y_{*}({\bf x})$ is constant inside the large sphere (to evolve into a man-made universe) and gradually aproaches zero as $r\to\infty$. It is straightforward to check that with an appropriate choice of the functions $F(Y;\varphi)$, $K(Y;\varphi)$, this construction does not encounter pathologies anywhere.

Let us now sketch a concrete construction. Let us assume that the field $\varphi$ is a usual scalar field which has two vacua, $\varphi=0$ and $\varphi=\varphi_{0}$. We prepare a spherical configuraion of this field with $\varphi=\varphi_{0}$ inside a sphere of large enough radius $R$ and $\varphi=0$ outside this sphere, see Fig. 3. We assume for definiteness that there is a source for the field $\varphi$ that keeps this configuration static. Let $L\ll R$ be the thickness of the wall separating the two vacua; $L$ is also kept time-independent by the source. We require that the mass of this ball is small enough, so that $R\gg R_{s}$, where $R_{s}$ is the Schwarzschild radius. The mass is of order $\mu^{4}R^{2}L$, where $\mu$ is the mass scale characteristic of the field $\varphi$. Hence, the latter requirement reads $\mu^{4}RL\ll M_{Pl}^{2}$. For small enough $\mu$ both $R$ and $L$ can be large.

Let the function $Y_{*}(\varphi)$ be such that $Y_{*}(0)=0$ and $Y_{*}(\varphi_{0})=Y_{0}$. We prepare the initial configuration of $\pi$ at $t=0$ in such a way that it initially evolves as

[TABLE]

where we allow the parameter $t_{*}$ in (34) to vary in space, and choose a convenient parametrization. We choose $t_{*}(r)=t_{*,in}$ inside a somewhat smaller sphere of radius $R_{1}<R$ (but $R_{1}\sim R$) and $t_{*}(r)=t_{*,out}\gg t_{*,in}$ at $r>R_{1}$ (hereafter subscripts in and out refer to the regions $r<R_{1}$ and $r>R_{1}$, respectively), as shown in Fig. 3, with the transition region of, say, the same thickness $L$. We take $t_{*,out}\ll L$, then the characteristic time scales are smaller than the smallest length scale $L$ inherent in the set up, so the spatial devivatives of $\pi$ are indeed negligible compared to the time derivatives. This ensures that the field $\pi$ is in the quasi-homogeneous regime. As $r\to\infty$, we have $Y_{*}(r)\to 0$ and $t_{*}\to\mbox{const}$, so the field $\pi$ tends to the Minkowski vacuum $\pi=\mbox{const}$.

At the initial stage of evolution, pressure inside the sphere of radius $R_{1}$ is

[TABLE]

where $M$ is the mass scale characteristic of the field $\pi$. We require that $p_{in}R^{3}/M_{Pl}^{2}\ll R$, then the gravitational potentials are small everywhere, and gravity is initially in the linear regime. Thus, we impose a constraint

[TABLE]

which is consistent with the above conditions for $M\ll M_{Pl}$ and $Y_{0}\gtrsim M^{2}$. In complete analogy with Section 4.2, the Hubble parameter inside the sphere of radius $R_{1}$ shortly after the beginning of evolution is

[TABLE]

In view of (59) and $t_{*,in}\ll R$, the Hubble length scale is large for some time, $H^{-1}\gg R$. This is true also at $r>R_{1}$, so there are no anti-trapped surfaces initially.

As $t$ approaches $t_{*,in}$, pressure in the Genesis region $r<R_{1}$ increases, and the Hubble length shrinks there to $R_{1}\sim R$. The anti-trapped surfaces get formed inside the sphere of radius $R_{1}$, a new universe gets created and enters the Genesis regime there. This occurs when $H_{in}\sim R^{-1}$, i.e., at time $t_{1}$ such that

[TABLE]

Note that at that time the energy density $\rho_{in}\sim M_{Pl}^{2}H_{in}^{2}$ is still relatively small,

[TABLE]

This implies that at time $t_{1}$, space-time is locally nearly Minkowskian. Another manifestation of this fact is that the scale factor is close to 1:

[TABLE]

where the correction to 1 is of order $\rho_{in}/p_{in}$. Hence, our approximate solution (58), (60) is legitimate.

Since $t_{*,out}\gg t_{*,in}$, the field $e^{\pi}$ at time $t_{1}$ is still small at $r>R_{1}$, and the Hubble length scale exceeds $R$ there. Gravity is still weak at $r>R_{1}$, so it is consistent to assume that the configuration of $\varphi$ is not modified by that time. Note also that a black hole is not formed by then either.

This completes the construction of the initial configuration and the analysis of the early epoch of a man-made universe. We make contact between this analysis and the general results of Ref. [7] in Appendix C.

Of course, the construction discussed here is merely a sketch. To make the scenario complete, one has to specify the way to design the configuration of the field $\varphi$ and keep it static (or consider an evolving field $\varphi$ instead). Also, one has to understand the role of spatial gradients. Finally, one would like to trace the dynamics of the system to longer times, with gravity effects included, and see what geometry develops towards the end of the Genesis epoch occuring at $r<R_{1}$. In particular, it is of interest to see whether a black hole gets formed.

5 Conclusion

The theories of (generalized) Galileons offer an interesting possibility of consistent and controllable NEC-violation. Still, there remain open issues. One of them is the danger of superluminality. While the background we consider in Section 4.1 may be safe in this respect, it is not impossible that other backgrounds are sick, especially when gravity generated by some other matter is relevant. An example of this sort is given in Ref. [93]. The superluminality issue is tightly related to the possibility of UV completion [53]. Another issue is the stability against the radiative corrections. While the simplest Galileon theories possess enough symmetries to guarantee the stability, generic Galileon Lagrangians (22) do not. There are also largely unexplored areas where NEC-violating theories may make surprizes, like black hole thermodynamics [94], absence/existence of closed time-like curves [95] and naked singularities [96], etc.

Of course, the most intriguing question is whether NEC-violating fields exist in Nature. Needless to say, no such fields have been discovered. The situation is not entirely hopeless, however: we may learn at some point in future that the Universe went through the bounce or Genesis epoch, and that will be an indication that NEC-violation indeed took place in the past.

The author is indebted to S. Demidov, D. Levkov, M. Libanov, I. Tkachev and M. Voloshin for helpful discussions and S. Deser, Y.-S. Piao and A. Vikman for useful correspondence. This work has been supported in part by the grant of the President of the Russian Federation NS-5590.2012.2 and the Ministry of Education and Science contract 8412.

6 Appendix A

Let us consider general spherically symmetric metric which we choose in diagonal form

[TABLE]

where $N=N(t,r)$, $a=a(t,r)$, $R=R(t,r)$. Our purpose is to show that trapped sphere is such that $R(t,r(t))$ decreases along outgoing null geodesic, for which $r$ increases.

The formal definition of a trapped sphere is that

[TABLE]

for a vector $l^{\mu}=dx^{\mu}/d\lambda$ tangent to outgoing radial null geodesic, where $\lambda$ is affine parameter. Vector $l^{\mu}$ is null,

[TABLE]

and obeys the geodesic equation

[TABLE]

For metric (62), eq. (63) gives

[TABLE]

where we have chosen to parametrise the geodesic by time $t$, so that the null world line is $(t,r(t),0,0)$; the sign of $l^{r}$ corresponds to outgoing geodesic. The normalization factor $u(t)$ is to be determined from eq. (64). To find this factor, we write $dl^{\mu}/d\lambda=dl^{\mu}/dt\cdot l^{0}$ and obtain the for the 0-th component of eq. (64):

[TABLE]

The relevant Christoffel symbols are

[TABLE]

where dot and prime denote partial derivatives, and Christoffel symbols entering eq. (65) are to be taken at $r=r(t)$. Thus, the function $u(t)$ obeys

[TABLE]

where, again, the terms in parenthesis are partial derivatives evaluated at $r=r(t)$. As a cross check, one writes the $r$-component of the geodesic equation (64),

[TABLE]

One makes use of

[TABLE]

and

[TABLE]

and finds that eq. (67) coincides with eq. (66).

We now calculate

[TABLE]

Using eq. (66) to eliminate $\dot{u}$, we arrive at

[TABLE]

Thus, the trapped surface is indeed such that $R(t,r(t))$ decreases along outgoing null geodesic.

As an example, for contracting spatially flat Universe we have $a=a(t)$, $R=a(t)r$, and the right hand side of eq. (68) is negative for $r>-1/\dot{a}$; a sphere of radius $R=ar>|H|^{-1}$ is trapped surface. By time reversal, a sphere of radius $R>|H|^{-1}$ in expanding Universe is anti-trapped surface.

7 Appendix B

Let us briefly discuss why the contracting Universe gets strongly inhomogeneous and anisotropic if the dominant matter obeys $p<\rho$, and why, on the contrary, it becomes more isotropic in the course of contraction in the opposite case. Let us consider a simplified version of anisotropic Universe which is descibed by the homogeneous anisotropic metric

[TABLE]

where $e_{i}^{(a)}$ are three linear independent vectors which are constant in time. We assume for simplicity that these vectors are orthogonal to each other (the dynamics is a lot more complicated in the general situation, but this turns out to be largely irrelevant from our viewpoint, see a comment below). The function $a(t)$ is chosen in such a way that

[TABLE]

in other words, $\mbox{det}g_{ij}=a^{6}$. The Einstein equations give

[TABLE]

Equation (70b) gives

[TABLE]

and in view of (69), the constants $d_{a}$ obey $\sum_{a}d_{a}=0$. Then Eq. (70a) becomes

[TABLE]

This equation shows that the overall contraction rate (the rate at which $\mbox{det}g_{ij}$ decreases) is determined at small $a$ by the anisotropy rather than matter, provided that $\rho$ increases slower than $a^{-6}$. For metric (7) covariant energy conservation still gives eq. (5) with $H=\dot{a}/a$, so the latter property holds for $p<\rho$. Therefore, one can set $\rho=0$ late at the collapsing stage, and the system of equations (71), (72) has the Kasner solution

[TABLE]

Hence, the anisotropy increases as the Universe collapses. In the general case when the vectors $e^{(a)}_{i}$ are not orthogonal to each other, this regime continues for finite time, and then the values of the parameters $d_{a}$ change in a rather abrupt manner [92]. The vectors $e^{(a)}_{i}$ change too. This change occurs infinitely many times in the limit $t\to 0$. This corresponds to the chaotic anisotropic collapse.

These results show that the Universe is very anisotropic before the bounce. In fact, the processes we described occur independently in Hubble-size regions and are very different in each of them because of their chaotic properties, so the Universe becomes strongly inhomogeneous too. This picture remains valid after the bounce, at least in the framework of the classical theory. Strong inhomogeneity of the Universe after the bounce is inconsistent with the smallness of the primordial cosmological perturbations, so the entire bounce scenario is up in the air.

To solve this problem, one invokes matter with super-stiff equation of state $p=w\rho$, $w>1$. Its energy density behaves as $\rho\propto a^{-3(1+w)}$, so it increases faster than $a^{-6}$. The second term in the right hand side of Eq. (72) dominates, the scale factor decreases as $a(t)\propto|t|^{\alpha}$ with $\alpha<1/3$, see eq. (49). It then follows from Eq. (71) that the parameters $\beta_{a}$ tend to constants as $t\to 0$. If the Universe is nearly homogeneous at the early stages of collapse, and anisotropy is not strong, then the Universe becomes more and more homogeneous in the process of contraction, see details in Ref. [91].

8 Appendix C

We show in this appendix that the results of Section 4.4 are in agreement with the general results of Ref. [7].

Definition [7]. Let the metric have the form (62). R-region is a region where normal vectors $R_{\mu}=\partial_{\mu}R$ to hypersurfaces $R=\mbox{const}$ are spacelike, $g^{\mu\nu}R_{\mu}R_{\nu}<0$. Since $g^{\mu\nu}R_{\mu}R_{\nu}=N^{-2}\dot{R}^{2}-a^{-2}R^{\prime 2}<0$, there is no place in an R-region where $R^{\prime}=0$, so the sign of $R^{\prime}$ is one and the same in the entire R-region. An R-region where $R^{\prime}>0$ is called $R_{+}$-region, while an R-region where $R^{\prime}<0$ is called $R_{-}$-region. T-region is a region where normal vectors $R_{\mu}$ to hypersurfaces $R=\mbox{const}$ are timelike, $g^{\mu\nu}R_{\mu}R_{\nu}>0$. There $\dot{R}$ is non-zero everywhere. Hence the sign of $\dot{R}$ is the same everywhere. A T-region where $\dot{R}>0$ is called $T_{+}$-region, while a T-region where $\dot{R}<0$ is called $T_{-}$-region. $T_{+}$- and $T_{-}$-regions are regions of expansion and contraction, respectively.

Let us consider the model of Section 4.4. In the above nomenclature, the whole space is initially $R_{+}$-region. At time $t_{1}$, a $T_{+}$-region appears. One of its boundaries moves towards smaller $r$, and another moves towards larger $r$. One of the results of Ref. [7] is that for $\rho+p<0$ ($\beta<0$ in nomenclature of Ref. [7]), the boundary between an inner $R_{+}$ region and $T_{+}$ region is necessarily space-like. Let us check that our geometry is consistent with this result.

In our case, $N=1$, $a\approx 1$ (see (61)) and $R=a(r,t)r$. The boundary between the left $R_{+}$-region and $T_{+}$-region is determined by $\dot{a}r=a$, i.e.,

[TABLE]

The normal to this hypersurface is the vector

[TABLE]

which is timelike, since

[TABLE]

Hence, the hypersurface separating the $R_{+}$- and $T_{+}$-regions is spacelike, in agreement with the general result of Ref. [7].

The outer boundary of the $T_{+}$-region may be in principle either spacelike or timelike [7]. For the same reason as above, it is actually spacelike in our case.