Cosmological bounce and Genesis beyond Horndeski

TL;DR

This paper presents a non-singular, stable bouncing cosmological model within beyond Horndeski theory, demonstrating a transition from a complex scalar field to a simple massless scalar at late times.

Contribution

It provides a specific stable bouncing solution in beyond Horndeski theory with a smooth transition to standard scalar field behavior.

Findings

The model is non-singular and stable throughout evolution.

The scalar field transitions from beyond Horndeski behavior to minimal coupling.

The solution demonstrates a viable alternative to Big Bang cosmology.

Abstract

We study a "classical" bouncing scenario in beyond Horndeski theory. We give an example of spatially flat bouncing solution that is non-singular and stable throughout the whole evolution. The model is arranged in such a way that the scalar field driving the cosmological evolution initially behaves like full-fledged beyond Horndeski, whereas at late times it becomes a massless scalar field minimally coupled to gravity.

Equations

$$S=\int {\mathrm{d}}^{4}x\sqrt{-g}\left({\mathcal{L}}_2+{\mathcal{L}}_3+{\mathcal{L}}_4+{\mathcal{L}}_5+{\mathcal{L}}_{\mathcal{B}\mathcal{H}}\right),$$

$${\mathcal{L}}_2=F(\pi ,X),$$

$${\mathcal{L}}_3=K(\pi ,X)\square \pi ,$$

$${\mathcal{L}}_4=-G_4(\pi ,X)R+2G_{4X}(\pi ,X)\left[{\left(\square \pi \right)}^2-{\pi }_{;\mu \nu }{\pi }^{;\mu \nu }\right],$$

$${\mathcal{L}}_5=G_5(\pi ,X)G^{\mu \nu }{\pi }_{;\mu \nu }+\frac{1}{3}{G}_{5X}\left[{\left(\square \pi \right)}^3-3\square \pi {\pi }_{;\mu \nu }{\pi }^{;\mu \nu }+2{\pi }_{;\mu \nu }{\pi }^{;\mu \rho }{\pi }_{;\rho }^{\nu }\right],$$

$${\mathcal{L}}_{\mathcal{B}\mathcal{H}}=F_4(\pi ,X){ϵ}_{\sigma }^{\mu \nu \rho }{ϵ}^{{\mu }^{\prime }{\nu }^{\prime }{\rho }^{\prime }\sigma }{\pi }_{,\mu }{\pi }_{,{\mu }^{\prime }}{\pi }_{;\nu {\nu }^{\prime }}{\pi }_{;\rho {\rho }^{\prime }}$$

$$+F_5(\pi ,X){ϵ}^{\mu \nu \rho \sigma }{ϵ}^{{\mu }^{\prime }{\nu }^{\prime }{\rho }^{\prime }{\sigma }^{\prime }}{\pi }_{,\mu }{\pi }_{,{\mu }^{\prime }}{\pi }_{;\nu {\nu }^{\prime }}{\pi }_{;\rho {\rho }^{\prime }}{\pi }_{;\sigma {\sigma }^{\prime }},$$

$$\mathrm{d}{s}^2=\mathrm{d}{t}^2-a^2(t){\delta }_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j.$$

$$\delta g^{00}:$$

$$-24H^{2}X(G_{4X}+G_{4XX}X)+12H{G}_{4\pi X}X\dot{\pi }$$

$$-2H^{3}X\dot{\pi }(5G_{5X}+2G_{5XX}X)+3H^{2}X(3G_{5\pi }+2G_{5\pi X}X)$$

$$+6H^2{X}^2(5F_4+2F_{4X}X)+6H^3{X}^2\dot{\pi }(7F_5+2F_{5X}X)=0,$$

$$\delta g^{ii}:$$

$$-8\dot{H}{G}_{4X}X-8H{G}_{4X}\ddot{\pi }\dot{\pi }-16H{G}_{4XX}X\ddot{\pi }\dot{\pi }$$

$$+2(\ddot{\pi }+2H\dot{\pi })G_{4\pi }+4X{G}_{4\pi X}(\ddot{\pi }-2H\dot{\pi })+2X{G}_{4\pi \pi }$$

$$-2X{G}_{5X}(2H^3\dot{\pi }+2H\dot{H}\dot{\pi }+3H^2\ddot{\pi })-4H^2{G}_{5XX}{X}^2\ddot{\pi }$$

$$+G_{5\pi }(3H^{2}X+2\dot{H}X+4H\ddot{\pi }\dot{\pi })+2H{G}_{5\pi X}X(2\ddot{\pi }\dot{\pi }-HX)$$

$$+2H{G}_{5\pi \pi }X\dot{\pi }+2F_{4}X(3H^{2}X+2\dot{H}X+8H\ddot{\pi }\dot{\pi })+8H{F}_{4X}{X}^2\ddot{\pi }\dot{\pi }$$

$$+4H{F}_{4\pi }{X}^2\dot{\pi }+6H{F}_5{X}^2(2H^2\dot{\pi }+2\dot{H}\dot{\pi }+5H\ddot{\pi })$$

$$+12H^2{F}_{5X}{X}^3\ddot{\pi }+6H^2{F}_{5\pi }{X}^3=0,$$

$$g_{00}=1+2\alpha ,g_{0i}=-{\partial }_i\beta ,g_{ij}=-a^2\left(e^{2\zeta }{\delta }_{ij}+h_{ij}^T\right),$$

$$\displaystyle S=\int\mathrm{d}t\mathrm{d}^{3}xa^{3}\Bigg{[}\left(\dfrac{\mathcal{\hat{G}_{T}}}{8}\left(\dot{h}^{T}_{ik}\right)^{2}-\dfrac{\mathcal{F_{T}}}{8a^{2}}\left(\partial_{i}h_{kl}^{T}\right)^{2}\right)+\Big{(}-3\mathcal{\hat{G}_{T}}\dot{\zeta}^{2}+\mathcal{F_{T}}\dfrac{(\triangledown\zeta)^{2}}{a^{2}}$$

$$\displaystyle-2\mathcal{G_{T}}\alpha\dfrac{\triangle\zeta}{a^{2}}+2\mathcal{\hat{G}_{T}}\dot{\zeta}\dfrac{\triangle\beta}{a^{2}}+6\Theta\alpha\dot{\zeta}-2\Theta\alpha\dfrac{\triangle\beta}{a^{2}}+\Sigma\alpha^{2}\Big{)}\Bigg{]},$$

$${\mathcal{G}}_{\mathcal{T}}=2G_4-4G_{4X}X+G_{5\pi }X-2H{G}_{5X}X\dot{\pi },$$

$${\mathcal{F}}_{\mathcal{T}}=2G_4-2G_{5X}X\ddot{\pi }-G_{5\pi }X,$$

$$\mathcal{D}=2F_{4}X\dot{\pi }+6H{F}_5{X}^2,$$

$${\hat{\mathcal{G}}}_{\mathcal{T}}={\mathcal{G}}_{\mathcal{T}}+\mathcal{D}\dot{\pi },$$

$$\Theta =-K_{X}X\dot{\pi }+2G_{4}H-8H{G}_{4X}X-8H{G}_{4XX}{X}^2+G_{4\pi }\dot{\pi }+2G_{4\pi X}X\dot{\pi }$$

$$-5H^2{G}_{5X}X\dot{\pi }-2H^2{G}_{5XX}{X}^2\dot{\pi }+3H{G}_{5\pi }X+2H{G}_{5\pi X}{X}^2$$

$$+10H{F}_4{X}^2+4H{F}_{4X}{X}^3+21H^2{F}_5{X}^2\dot{\pi }+6H^2{F}_{5X}{X}^3\dot{\pi },$$

$$\Sigma =F_{X}X+2F_{XX}{X}^2+12H{K}_{X}X\dot{\pi }+6H{K}_{XX}{X}^2\dot{\pi }-K_{\pi }X-K_{\pi X}{X}^2$$

$$-6H^2{G}_4+42H^2{G}_{4X}X+96H^2{G}_{4XX}{X}^2+24H^2{G}_{4XXX}{X}^3$$

$$-6H{G}_{4\pi }\dot{\pi }-30H{G}_{4\pi X}X\dot{\pi }-12H{G}_{4\pi XX}{X}^2\dot{\pi }+30H^3{G}_{5X}X\dot{\pi }$$

$$+26H^3{G}_{5XX}{X}^2\dot{\pi }+4H^3{G}_{5XXX}{X}^3\dot{\pi }-18H^2{G}_{5\pi }X-27H^2{G}_{5\pi X}{X}^2$$

$$-6H^2{G}_{5\pi XX}{X}^3-90H^2{F}_4{X}^2-78H^2{F}_{4X}{X}^3-12H^2{F}_{4XX}{X}^4$$

$$-168H^3{F}_5{X}^2\dot{\pi }-102H^3{F}_{5X}{X}^3\dot{\pi }-12H^3{F}_{5XX}{X}^4\dot{\pi }.$$

$$\frac{△\beta }{a^2}:$$

$$\alpha :$$

$$S=\int \mathrm{d}t{\mathrm{d}}^{3}x{a}^3\left[\frac{{\hat{\mathcal{G}}}_{\mathcal{T}}}{8}{\left({\dot{h}}_{ik}^T\right)}^2-\frac{{\mathcal{F}}_{\mathcal{T}}}{8a^2}{\left({\partial }_i{h}_{kl}^T\right)}^2+{\mathcal{G}}_{\mathcal{S}}{\dot{\zeta }}^2-{\mathcal{F}}_{\mathcal{S}}\frac{(▽\zeta {)}^2}{a^2}\right],$$

$${\mathcal{G}}_{\mathcal{S}}=\frac{\Sigma {{\hat{\mathcal{G}}}_{\mathcal{T}}}^2}{{\Theta }^2}+3{\hat{\mathcal{G}}}_{\mathcal{T}},$$

$${\mathcal{F}}_{\mathcal{S}}=\frac{1}{a}\frac{\mathrm{d}\xi }{\mathrm{d}t}-{\mathcal{F}}_{\mathcal{T}},$$

$$\xi =\frac{a{\mathcal{G}}_{\mathcal{T}}{\hat{\mathcal{G}}}_{\mathcal{T}}}{\Theta }=\frac{a\left({\hat{\mathcal{G}}}_{\mathcal{T}}-\mathcal{D}\dot{\pi }\right){\hat{\mathcal{G}}}_{\mathcal{T}}}{\Theta }.$$

$$c_{\mathcal{T}}^2=\frac{{\mathcal{F}}_{\mathcal{T}}}{{\hat{\mathcal{G}}}_{\mathcal{T}}},c_{\mathcal{S}}^2=\frac{{\mathcal{F}}_{\mathcal{S}}}{{\mathcal{G}}_{\mathcal{S}}}.$$

$${\hat{\mathcal{G}}}_{\mathcal{T}}>{\mathcal{F}}_{\mathcal{T}}>0,{\mathcal{G}}_{\mathcal{S}}>{\mathcal{F}}_{\mathcal{S}}>0.$$

$$\xi (t_2)-\xi (t_1)=\underset{t_1}{\overset{t_2}{\int }}a(t)\left({\mathcal{F}}_{\mathcal{T}}+{\mathcal{F}}_{\mathcal{S}}\right)\mathrm{d}t.$$

$$\xi (t_1)=\xi (t_2)-\underset{t_1}{\overset{t_2}{\int }}a(t)\left({\mathcal{F}}_{\mathcal{T}}+{\mathcal{F}}_{\mathcal{S}}\right)\mathrm{d}t.$$

$$\xi (t_2)=-|\xi (t_1)|+\underset{t_1}{\overset{t_2}{\int }}a(t)\left({\mathcal{F}}_{\mathcal{T}}+{\mathcal{F}}_{\mathcal{S}}\right)\mathrm{d}t,$$

Full text

1 Introduction

Bouncing cosmologies are interesting scenarios that are complementary or alternative to inflation. These models enable one to solve the singularity problem, and at the same time possess many nice features of inflationary theories.

A model of spatially flat bounce requires the violation of the Null Energy Condition (NEC), which may be realised by some fairly unconventional matter. One of the possible candidates for the latter are the generalized Galileons, that are equivalent to Horndeski theory [1, 2, 3, 4, 5, 6, 7, 8]. Galileons are scalar fields, whose Lagrangians involve second derivatives but the corresponding field equations are still of the second order (for a review see, e.g. [9]).

It was shown, that generalized Galileons can indeed be utilized for constructing bouncing Universe models, which have no pathologies (ghosts, gradient instabilities, etc.) around the moment of bounce [10, 11, 12, 13]. However, the issue of avoiding instabilities becomes more challenging if one considers more complete models. Indeed, with generalized Galileons, when trying to extend pathology-free bounce solutions to long enough periods of time, one always meets gradient (or, possibly, ghost) instabilities [14, 15, 16, 17, 18, 19, 20]. Similar observations apply to (“complete”) Genesis models [18, 22].

One of the possible ways to get around the gradient instability problem is to arrange a model in such a way that the term in the action, which is quadratic in spatial gradient and has the wrong sign, is small so that higher derivative terms restore stability at sufficiently high momenta [21, 22]. Another possibility is that the strong coupling momentum scale is low enough [23]. In both cases the exponential growth of perturbations does not lead to a catastrophic outcome, provided that the cosmic time interval when the instabilities are present, is short.

On the other hand, it is definitely of interest whether geodesically complete, healthy cosmological models — the flat bouncing scenario and Genesis in particular — can be constructed on the basis of the generalized Galileons. The inevitable presence of pathologies within a subclass of Horndeski theories was formulated as a no-go theorem [24] and further generalised to the case of direct interaction of the Galileon field with conventional scalar field [25]. The further extension of the no-go argument was presented in [26], showing that complete cosmological models, based on the most general Horndeski theories, are always plagued with instabilities. This no-go theorem has been further generalized to a multi Galileon case [27]. There was a loophole, though, mentioned in [26] in the Genesis context: it is possible to construct non-pathological bouncing or Genesis solution, in which strong coupling gravity regime occurs long before or after the bouncing stage [28].

Recently, it has been proposed that Horndeski theories can be safely extended [29]. Even though the newly introduced Lagrangian terms lead in general to the presence of third derivatives in the equations of motion, the theory still does not contain extra degrees of freedom. These new terms in the Lagrangian may hopefully be utilized for solving the problem of instabilities in the complete cosmologies. This expectation has been strongly substantiated within the Effective Field Theory (EFT) approach [31, 30], see Sec. 2 for further discussion.

Our purpose in this paper is to present the full Lagrangians of the models, belonging to the class of beyond Horndeski theory, and construct classical bouncing and Genesis solutions which are non-singular during the entire evolution. Our set up does not have any kind of fine-tuning, nor it gets into strongly coupled gravity regime. The gradient and/or ghost instabilities are absent about our backgrounds at any time. We also address the issue of scale covariance at early times, which is behind the simple form of our early-time solution. In the Genesis case, we explicitly show the geodesic completeness of our solution.

The paper is organized as follows. In Sec. 2 we give the general Lagrangian of beyond Horndeski theory and obtain the quadratic action for perturbations in the unitary gauge. We provide a recipe for getting around the no-go argument existing in the generalized Galileon theories. We construct a healthy bouncing solution in Sec. 3 and give an analysis of its stability. There we also discuss the issue of early-time scale covariance of the theory. In Sec. 4 an example of healthy and geodesically complete Genesis is described. We conclude in Sec. 5.

Note added: while this paper was close to completion, a preprint [32] appeared with different construction for stable classical bounce in beyond Horndeski theory.

2 Stability of solutions in beyond Horndeski model

We consider beyond Horndeski theory with the following action:

[TABLE]

where $\pi$ is the Galileon field, $\pi_{,\mu}=\partial_{\mu}\pi$, $X=g^{\mu\nu}\pi_{,\mu}\pi_{,\nu}$, $\pi_{;\mu\nu}=\triangledown_{\nu}\triangledown_{\mu}\pi$, $\Box\pi=g^{\mu\nu}\triangledown_{\nu}\triangledown_{\mu}\pi$, $G_{4X}=\partial G_{4}/\partial X$, etc. Horndeski theory is obtained if one sets $F_{4}(\pi,X)=F_{5}(\pi,X)=0$. It is worth noting that action (1) already has gravitational parts, furthermore, one can restore the Einstein-Hilbert gravity by choosing $G_{4}(\pi,X)=\frac{1}{2\kappa}$, $G_{5}(\pi,X)=0$, where $\kappa=8\pi G$ and $G$ is the gravitational constant. We start from the unperturbed spatially flat FLRW metric (mostly negative signature)

[TABLE]

Variation of action (1) with respect to $g^{00}$ and $g^{ii}$ leads to the equations of motion

[TABLE]

where dot denotes the derivative with respect to $t$ and $H=\dot{a}/a$ is the Hubble parameter.

We study Lagrangian for perturbations in the unitary gauge, which means that we choose perturbations of the scalar field equal to zero, $\delta\pi=0$. We adopt the following parametrization:

[TABLE]

where $h_{ij}^{T}$ denotes tensor perturbations: $h^{T}_{ii}=0,\quad\partial_{i}h^{T}_{ij}=0$. Our concern is the stability against the most rapidly developing perturbations, which is determined by the UV behavior of the theory. Therefore, we only keep terms with the highest order of spatial and temporal derivatives and the Lagrange multipliers. Quadratic action for perturbations then reads:

[TABLE]

with the following coefficients:

[TABLE]

The key property specific to beyond Horndeski theory as compared to Horndeski is the difference between $\mathcal{\hat{G}_{T}}$ and $\mathcal{G_{T}}$, which comes precisely from beyond Horndeski term $\dot{\pi}\mathcal{D}$. This new term will allow us to construct a stable beyond Horndeski bounce.

Variation of action (3) with respect to the Lagrange multipliers leads to the constraint equations:

[TABLE]

Substituting these into action (3) we obtain quadratic action in the unconstrained form (similar expression has been obtained in ADM formalism in Ref. [29])

[TABLE]

where the coefficients are

[TABLE]

The speeds of sound for tensor and scalar perturbations are, respectively,

[TABLE]

A healthy and stable solution requires correct signs for kinetic and gradient terms as well as subluminal propagation:

[TABLE]

Horndeski theories are known to prohibit the stable bounce. Let us briefly summarize the proof of the corresponding no-go theorem and show how to bypass it by introducing beyond Horndeski terms. We will largely follow the proof given in Refs. [24], [26]. The proof starts with Eq. (7b) expressed in an integrated form:

[TABLE]

For a bounce, the scale factor initially decreases, reaches minimum and then starts increasing. Thus, it is bounded by its minimal value at the moment of bounce $a(t)\geq a_{min}>0$. Furthermore, we want to keep $\mathcal{F_{T}}$ and $\mathcal{F_{S}}$ positive and bounded from below, otherwise we are back to the option of the domination of higher order spatial derivatives [21]. This means that the combination $a(t)(\mathcal{F_{T}}+\mathcal{F_{S}})$ is always positive and bounded from below. The same argument applies to geodesically complete bounce [31].

Now, suppose that $\xi(t_{2})>0$. We have

[TABLE]

This shows that at early enough times $t_{1}$, one has $\xi(t_{1})<0$. Another possibility is that $\xi(t_{1})<0$. Then we write

[TABLE]

and at large enough $t_{2}$ one has $\xi(t_{2})>0$.

Hence, there must be a moment of time when $\xi(t)$ changes sign, i.e, it crosses zero, $\xi(t_{0})=0$. This property is valid both in Horndeski and beyond Horndeski theories. In the Horndeski case $\mathcal{\hat{G}_{T}}=\mathcal{G_{T}}$, and, therefore, Eq. (7c) reads $\xi=\frac{a\mathcal{G_{T}}^{2}}{\Theta}$. Hence, zero crossing for $\xi$ requires $\Theta\rightarrow\infty$, which means a singularity of Lagrangian functions in action (1).111Another option is to let $\mathcal{G_{T}}$ and $\Theta$ become zero at the time when $\xi=0$  [28]. At least naively, the theory becomes strongly coupled near $\xi=0$ in this case. Yet another possibility is that $\mathcal{F_{T}}\rightarrow 0$ and $\mathcal{F_{S}}\rightarrow 0$ as $t\rightarrow-\infty$ [26, 28], so that the integral (9) is convergent in the lower limit. This again brings one to a strong gravity regime.

In contrast to Horndeski theory, the expression for $\xi$ beyond Horndeski involves $(\mathcal{\hat{G}_{T}}\cdot\mathcal{G_{T}})$, rather than $\mathcal{{G}_{T}}^{2}$. $\mathcal{G_{T}}$ is no longer a physically important kinetic coefficient, so it can cross zero and be negative at large negative times. At the same time, with the extra term $\dot{\pi}D$ in (4c) it is possible to have $\mathcal{\hat{G}_{T}}$ positive at all times. Hence, the difference between $\mathcal{G_{T}}$ and $\mathcal{\hat{G}_{T}}$ enables one to have regular $\xi=0$ point with positive $\mathcal{\hat{G}_{T}}$ everywhere. This can be achieved only as a joint play of $\mathcal{L}_{4}$ or $\mathcal{L}_{5}$ Lagrangians together with $\mathcal{L_{BH}}$, see eqs. (4a) and (4c).

We note that the same arguments apply to the Genesis case. Namely, healthy Genesis is impossible without introducing beyond Horndeski terms (modulo “loopholes” related to strong gravity regime and geodesic incompleteness [24, 26]). On the other hand, the difference between $\mathcal{\hat{G}_{T}}$ and $\mathcal{G_{T}}$ suggests that healthy Genesis may be possible in beyond Horndeski theory.

The quadratic action for perturbations in beyond Horndeski theory was also obtained within the EFT approach [30, 31], where a similar set of coefficients was found. The correspondence between the coefficients in eqs. (4) and (7) and ones given in Ref. [31] is

[TABLE]

Similar relations can be obtained for Ref. [30].

Refs. [30] and [31] proposed the same idea to evade the no-go argument as we have described above. Namely, the beyond Horndeski term $\bar{m}_{3}^{2}$ (which equals to $\dot{\pi}\mathcal{D}$ in our language) was used to allow $Y$ ($\xi$ in our case) to safely cross zero, while ensuring positive kinetic coefficient $M_{Pl}^{2}$ ($\mathcal{\hat{G}_{T}}$ in our case). However, the EFT approach does not enable one to write the corresponding 4D Lagrangian (1) as well as to check whether the equations of motion (2) are satisfied.

Before we jump to a specific example of a healthy beyond Horndeski bounce, there is an issue to discuss. Let us consider the case of $\Theta$ crossing zero at some moment $t_{0}$. In order to have regular $\mathcal{F_{S}}$ and $\mathcal{G_{S}}$ one would have to fine tune the coefficients $\Sigma$ and $\mathcal{G_{T}}$ (see eqs. (7)). The numerators in $\mathcal{F_{S}}$ and $\mathcal{G_{S}}$ have to have higher order zeros at the same moment of time $t_{0}$. It is this case that was considered when constructing the example for a complete bounce in Ref. [31].

We are going to avoid this fine tuning and keep $\Theta\neq 0$ at all times. This comes with a price: the Galileon field $\pi$ cannot be conventional scalar field, described solely by the Lagrangian $\mathcal{L}_{2}$ and the Einstein–Hilbert term, in both distant past and future. Otherwise, Eq. (4e) states that such conventional scalar field has $\Theta$ proportional to the Hubble parameter, $\Theta=2G_{4}H$, where $G_{4}$ is the gravitational constant; the bounce starts with $H<0$ and ends with $H>0$, therefore, $\Theta$ must change sign and cross zero at some point. On the contrary, in this paper we are going to construct the bouncing scenario with non-trivial Galileon field $\pi$ in the asymptotic past, which eventually evolves into a conventional scalar field in distant future. In this way we keep $\Theta$ finite and positive at all times.

It is impossible also to cook up a Genesis model where the Galileon is massless scalar field both at late and early times. The argument is slightly more involved as compared to the bounce case. It goes as follows. We want to have Einstein–Hilbert gravity at distant past, so we impose at early times

[TABLE]

Thus, at early times one has

[TABLE]

However, according to the mechanism of avoiding the no-go theorem, $\mathcal{G_{T}}$ must change sign at some point and is negative at early times, in contradiction with (10). There is a possible loophole: the latter argument assumes that $\mathcal{G_{T}}$ crosses zero only once. To see that this is the case we recall that

[TABLE]

so the function $\xi$ is always growing. Therefore, it can cross zero only once. According to our discussion above, $\mathcal{G_{T}}$ crosses zero at the same moment as $\xi$, thus $\mathcal{G_{T}}$ crosses zero only once as well. This completes the argument.

Note that the same argument works in the case of bounce. It does not rely upon the behavior of $\Theta$ and irrespectively of this behavior shows that it is impossible to cook up a bouncing model where the Galileon is massless scalar field both at late and early times.

3 Cosmological bounce: an example

In this section we give a concrete example of the Lagrangian functions and solution to the field equations which describes a classical, fully stable cosmological bounce. We perform the analysis in Planck units and set

[TABLE]

Of course, upon introducing an appropriate dimensionless parameter in the Lagrangian functions, the resulting dynamics can be made safely sub-Planckian.

Before we choose an explicit form of the Lagrangian functions, let us recall the basic requirement in our scenario. At late times, we require the Galileon to become a conventional massless scalar field, whose equation of state is $p=\rho$, where $p$ is the pressure and $\rho$ is the energy density. Then, the late-time asymptotic of the Hubble parameter is

[TABLE]

Our purpose is to find the Lagrangian functions such that the model admits healthy bounce solution. To this end, we choose the time dependence of some of the background functions at our will, and reconstruct other functions by solving the field equations (so, we adopt a general approach advocated in Refs. [24, 26, 20]). In particular, we choose the Hubble parameter at all times equal to

[TABLE]

so the bounce occurs at $t=0$. For the scale factor $a(t)$ one has

[TABLE]

The evolution of the Hubble parameter and its time derivative are shown in Fig. 1.

Let us choose the Lagrangian functions as follows:

[TABLE]

Note that it is sufficient for our purposes to set $G_{5}=F_{5}=0$. We make a field redefinition in such a way that that the background scalar field rolls with constant speed:

[TABLE]

We immediately obtain that $X=\left(\partial\pi\right)^{2}=1$. For derivatives of the Lagrangian functions one has

[TABLE]

Our implementation of the approach outlined above is as follows: we first specify the explicit forms of $k_{1}(t)$, $g_{40}(t)$, $g_{41}(t)$ and $f_{40}(t)$ at all times. Then we express the functions $f_{1}(t)$ and $f_{2}(t)$ through $k_{1}(t)$, $g_{40}(t)$, $g_{41}(t)$ and $f_{40}(t)$ via the Einstein equations, again at all times. The most tricky thing is then to specify the remaining function $f_{0}(t)$ in such a way that the conditions of ghost absence and subluminality are satisfied.

We now turn to functions $k_{1}(t)$, $g_{40}(t)$, $g_{41}(t)$ and $f_{40}(t)$. Since at late times, the resulting Lagrangian should correspond to a theory of free massless scalar field and Einstein-Hilbert gravity, it follows from Eq. (1) that we should have the following late-time asymptotics of $K$, $G_{4}$ and $F_{4}$:

[TABLE]

On the other hand, as we discussed in the previous section, it is impossible to cook up a scenario with a massless scalar field both at late and early times. So, the early-time behavior of our system is less constrained. One of the possible (and pretty arbitrary) options for early- time asymptotics of $G_{4}$ and $F_{4}$ is

[TABLE]

In contrast, a convenient choice for $K$ is that it is not equal to constant at early times. Since we have defined asymptotics of $G_{4}$, $F_{4}$ and Hubble parameter, it is natural to conjecture the behavior of $K$ by considering expression for $\Theta$, because it includes all these functions and does not include $F$. From (4e) and (13) one has

[TABLE]

Considering the latter equation at early times and taking into account Eq. (16), one concludes that $k_{1}(t)$ is naturally chosen proportional to $H(t)$ to have a simple power-law behavior of $\Theta$.

Taking into account asymptotics of Lagrangian functions (15), (16) and the fact that $K$ should be proportional to Hubble parameter at early times, let us choose $k_{1}(t)$, $g_{40}(t)$, $g_{41}(t)$ and $f_{40}(t)$ in the following form

[TABLE]

where $c_{1}$, $c_{2}$, $c_{3}$ and $c_{4}$ are constant coefficients.

Our main purpose now is to find time dependence of $f_{0}(t)$, $f_{1}(t)$ and $f_{2}(t)$. Note that the only non-vanishing term in the Lagrangian at late times is $f_{1}(t)$, since it corresponds to conventional kinetic term, and the Lagrangian describes a theory of massless scalar field in this case. Thus, $f_{0}(t)$ and $f_{2}(t)$ vanish at late times

[TABLE]

We define $f_{1}(t)$ as a sum of two functions

[TABLE]

with the following late-time asymptotics:

[TABLE]

where $p$ is a constant coefficient. Let us consider asymptotic behavior of Einstein equations (2) at $t\rightarrow+\infty$. The only non-vanishing terms are $H(t)$, $\dot{H}(t)$ and $f_{10}(t)$. By substituting $H(t)=1/3t$ and $f_{10}(t)=p/t^{2}$ into Eq. (2) with all other Lagrangian functions equal to zero one finds that $p=1/3$, which is a standard result in the gauge $\pi=t$. So, we choose $f_{10}(t)$ equal to

[TABLE]

at all times. Note that $f_{11}(t)$ remains undefined, modulo its rapid decay as $t\rightarrow+\infty$, see (19).

Let us now consider Einstein equations at all times. Substituting eqs. (13) and (18) into (2) and solving for $f_{11}(t)$ and $f_{2}(t)$, we find

[TABLE]

Now, it is not immediate to choose the remaining undefined function $f_{0}(t)$ to satisfy the conditions of ghost absence and subluminality. As an intermediate step, we require that $f_{0}(t)$ satisfies the equation

[TABLE]

where $\Sigma$ is determined by (4f) and $q$ is a constant coefficient. From (7) and (22) one has

[TABLE]

From the latter equation it follows that one can vary $\mathcal{G_{S}}$ by changing coefficient $q$ and, therefore, by changing $f_{0}(t)$. We will make use of latter equation later. Solving Eq. (22) for $f_{0}(t)$, one has

[TABLE]

By making use of eqs. (11), (17), (20), (21) and (24) one can check that $f_{0}(t)$, $f_{11}(t)$ and $f_{2}(t)$ are smooth functions that rapidly vanish at late times. The Lagrangian will be fully defined if we specify the values of coefficients $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$ and $q$. Note that we have not yet established the absence of instabilities.

Before we turn to the stability conditions of scalar and tensor perturbations, let us recall that we want positive $\Theta$ at all times. With chosen asymptotics of Lagrangian functions, one has $\Theta|_{t=+\infty}=H$ at late times. The theory describes Einstein–Hilbert gravity and free massless scalar field at late times, therefore, $G_{4}(\pi,X)=\frac{1}{2}$, and $\mathcal{G_{T}}|_{t=+\infty}=1$. At early times, we require

[TABLE]

Now, recall that we want $\mathcal{G_{T}}$ to turn sign at some point; thus, we require

[TABLE]

Let us now turn to the stability conditions (8).

Consider stability conditions for tensor perturbations. In Einstein-Hilbert gravity at late times, one has

[TABLE]

We require $\mathcal{\hat{G}_{T}}$ and $\mathcal{F_{T}}$ to be smooth monotonic functions of $t$ and satisfy the following inequalities at early times

[TABLE]

If eqs. (27) and (28) are valid, then the smooth monotonic functions $\mathcal{\hat{G}_{T}}$ and $\mathcal{F_{T}}$ are positive at all times. In addition, we will choose them in such a way that inequality $\mathcal{\hat{G}_{T}}>\mathcal{F_{T}}$ is satisfied at all times as well. 2. 2.

We focus now on the stability conditions for scalar perturbations. One can check that inequality $\mathcal{F_{S}}|_{t=+\infty}>0$ is equivalent to $\dot{H}|_{t=+\infty}<0$ and is satisfied with chosen Hubble parameter (11). At early times, we require

[TABLE]

Let us leave aside for a moment the conditions of ghost absence $\mathcal{G_{S}}>0$ and subluminality $\mathcal{G_{S}}>\mathcal{F_{S}}$. One possible choice of coefficients $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$, such that the inequalities (25), (26), (28) and (29) are valid, is

[TABLE]

Now we recall Eq. (23) from which it follows that the condition of ghost absence $\mathcal{G_{S}}>0$ is automatically satisfied when $\mathcal{\hat{G}_{T}}>0$. Moreover, by taking

[TABLE]

the condition $\mathcal{G_{S}}>\mathcal{F_{S}}$ is satisfied as well.

Thus, with parameters (30), (31) we have a completely healthy bounce in our beyond Horndeski theory. Importantly, inequalities (25), (26), (28) and (29) with sufficiently large $q$ guarantee the absence of pathologies only at early and late times. However, with the coefficients $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$ and $q$ given by (30) and (31), dynamics is stable throughout the entire evolution. The behavior of scalar and tensor kinetic and gradient terms is shown in Figs. 2 and 3.

Both $\xi$ and $\mathcal{G_{T}}$ change sign simultaneously at $\pi=t=\operatorname{artanh}\left(\frac{7}{9}\right)\approx-1$, but $\mathcal{\hat{G}_{T}}$ stays positive at all times, and our mechanism of evading the no-go theorem works. The behavior of $\xi$ and $\mathcal{G_{T}}$ is shown in Fig. 4.

Now let us turn to the Lagrangian functions. From eqs. (11), (13), (14), (17), (18), (20), (21) and (24) the Lagranginan functions have the following asymptotics at early times (we no longer use the gauge $\dot{\pi}=1$):

[TABLE]

where the coefficients $\mathcal{C}_{0}$, $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are

[TABLE]

These can be found from eqs. (21) and (24) at $t\rightarrow-\infty$. The Lagrangian at early times has the form

[TABLE]

Note that it contains both Horndeski and beyond Horndeski terms. On the other hand, at late times the only non-vanishing function is $f_{1}(t)$. From eqs. (14), and (20) at late times one has

[TABLE]

The Lagrangian at $t=+\infty$ has the form

[TABLE]

where $\phi=\ln(\pi)$. As we anticipated, Galileon field becomes a free massless scalar field interacting with the Einstein-Hilbert gravity.

To discuss the overall behavior of the Lagrangian functions, we recall that $k_{1}(\pi)$, $g_{40}(\pi)$, $g_{41}(\pi)$, and $f_{40}(\pi)$ are explicitly given by (17), with $t$ substituted by $\pi$ (we are back to the gauge $\dot{\pi}=1$). The rest of the functions (namely, $f_{0}$, $f_{1}$, $f_{2}$) are shown in Fig. 5, where constant coefficients are chosen as in eqs. (30) and (31).

The theory possesses one peculiar feature: it is scale covariant at $t=-\infty$. This scale covariance is behind the simple asymptotic power-law behavior at early times. Let us make a scale transformation of the coordinates

[TABLE]

The Galileon field $\pi(x^{\mu})$ and metric $g_{\mu\nu}(x^{\mu})$ transform as follows:

[TABLE]

and, respectively, $g^{\prime\mu\nu}(x^{\mu})=\lambda g^{\mu\nu}(\lambda x^{\mu})$. One can check that the early time Lagrangian transforms in the following way:

[TABLE]

and the equations of motion remain unchanged. Moreover, the solution is scale invariant in terms of conformal time $\eta=\int\frac{\mathrm{d}t}{a(t)}$. Indeed, as $t\rightarrow-\infty$, the asymptotics of the solution is

[TABLE]

where $\eta_{\mu\nu}$ is the Minkowski metric. From eqs. (36) and (37) it follows that the solution is invariant under the transformations (35) at early times.

4 Genesis: an example

We now apply the same technique to construct stable, geodesically complete Genesis solution. The only thing that we have to change as compared to the previous reasoning is the Hubble parameter and function $k_{1}(t)$, since the latter is proportional to the Hubble parameter at early times.

Like in the case of the bounce, we require the Galileon to become a conventional massless scalar field at late times. Thus, the late-time asymptotic of the Hubble parameter is

[TABLE]

We choose the Hubble parameter at all times equal to

[TABLE]

For the scale factor $a(t)$ one has

[TABLE]

The evolution of the Hubble parameter and its time derivative are shown in Fig. 6.

As we have argued earlier, we only have to change $k_{1}(t)$. In the case of bounce it was chosen to be proportional to the Hubble parameter at early times, and to vanish at late times. We make a similar Ansatz in the Genesis case, and choose $k_{1}(t)$ in the following form:

[TABLE]

The Ansatz for the the functions $g_{40}$, $g_{41}$ and $f_{40}$ is completely the same as in (17b), (17c), and we again request that Eq. (22) is satisfied. Then the functions $f_{11}$, $f_{2}$ and $f_{0}$ are again found from (21), (24). It remains to make a choice of constants; our example is

[TABLE]

and

[TABLE]

With this choice, the Genesis solution is completely healthy. The evolution of kinetic and gradient terms as well as the speed of scalar and tensor perturbations are presented in Fig. 7.

To complete our discussion of the Genesis solution, we show in Fig. 8 the behavior of $\xi$ and $\mathcal{G_{T}}$, which enables one to evade the no-go argument, cf. Fig. 4. The functions $\xi(t)$ and $\mathcal{G_{T}}$ cross zero at $\pi=t=\operatorname{artanh}\left[\frac{1}{17}\right]\approx 0.06$. The functions $f_{0}$, $f_{1}$, $f_{2}$ are shown in Fig. 9, where the constant coefficients are chosen as in eqs. (38) and (39). We see that the Genesis solution is completely healthy, with all Lagranian functions smooth at all values of $\pi$.

Finally, let us consider asymptotic completeness of our Genesis solution. Since the asymptotics of the Lagrangian functions are similar to those in Sec. 3, our model of Genesis is also scale covariant at $t\rightarrow-\infty$. Recall now that the scale factor for our scenario has the form

[TABLE]

At early times ($t\rightarrow-\infty$), the scale factor has the following asymptotics:

[TABLE]

Let us consider the null and timelike geodesics

[TABLE]

where prime denotes the derivative with respect to affine parameter $\tau$. We keep only one coordinate $x$, since the geodesic is a straight line. One of the two geodesic equations has the form

[TABLE]

which implies that

[TABLE]

where we can set $C=1$ without loss of generality. From the latter equation and eqs. (41) one obtains the following expressions for the null geodesic

[TABLE]

and the timelike one

[TABLE]

We see that (43a) and (43b) have same asymptotic behavior as (42a) and (42b), since $a(t)\rightarrow 0$ and, therefore, $\sqrt{1+a(t)^{2}}\rightarrow 1$ for $t\rightarrow-\infty$. Hence, we can study a simpler set of equations (42a) and (42b). From eqs. (40), (42a) and (42b) we get

[TABLE]

Moving to the past along the geodesic ($t\rightarrow-\infty$), the affine parameter tends to infinity ($\tau\rightarrow-\infty$), and so does the coordinate ($x\rightarrow-\infty$). Hence, at no value of the affine parameter we leave the FRWL coordinate patch. Thus, our Genesis model is geodesically complete.

5 Conclusion

In this paper we constructed an explicit “classical” cosmological bounce that is free of any kind of instabilities and singularities during the whole evolution. We also gave an example of fully stable and geodesically complete Genesis. We presented the Lagrangian functions and checked that the Einstein and field equations are satisfied. We also found that the simple form of the solution at early times is the result of the asymptotic scale covariance of the theory. The characteristic feature of the solutions, namely, the flow of the Galileon field into a conventional massless scalar field at late times, enables one to potentially merge the bouncing and/or Genesis scenario with the conventional evolution at later stages.

Acknowledgements

We are indebted to V. Rubakov for fruitful discussions and thoughtful reading of the manuscript. This work has been supported by Russian Science Foundation grant 14-22-00161.