Galilean Creation of the Inflationary Universe

It has been pointed out that the null energy condition can be violated stably in some non-canonical scalar-field theories. This allows us to consider the Galilean Genesis scenario in which the universe starts expanding from Minkowski spacetime and hence is free from the initial singularity. We use this scenario to study the early-time completion of inflation, pushing forward the recent idea of Pirtskhalava et al. We present a generic form of the Lagrangian governing the background and perturbation dynamics in the Genesis phase, the subsequent inflationary phase, and the graceful exit from inflation, as opposed to employing the effective field theory approach. Our Lagrangian belongs to a more general class of scalar-tensor theories than the Horndeski theory and Gleyzes-Langlois-Piazza-Vernizzi generalization, but still has the same number of the propagating degrees of freedom, and thus can avoid Ostrogradski instabilities. We investigate the generation and evolution of primordial perturbations in this scenario and show that one can indeed construct a stable model of inflation preceded by (generalized) Galilean Genesis.


I. INTRODUCTION
Inflation in the early Universe [1,2] is now an indispensable ingredient of modern cosmology not only to explain the global properties of homogeneous and isotropic space with a vanishingly small spatial curvature but also to account for the origin of the primordial curvature perturbation that seeded cosmic structure formation [3]. At present, despite the significant progress in the state-of-the-art precise measurements of the cosmic microwave background radiation (CMB) by WMAP [4,5] and Planck [6,7] missions, there is no single observational result in conflict with the single-field inflation paradigm [2]. In particular, the anti-correlation of the temperature and the E-mode polarization anisotropies on large scales observed by the WMAP mission strongly supports the superhorizon perturbations suggested by inflation [8].
In other words, once inflation sets in, virtually all the available cosmological observation data can be explained simultaneously irrespective of the initial condition of the Universe. This does not mean that we may be indifferent to the initial condition of the Universe before inflation. On the contrary, in order to achieve complete understanding of the cosmic history, we must work out the very beginning of the Universe that may smoothly evolve into the inflationary phase.
As is well known, as long as the null energy condi- * Email: tsutomu"at"rikkyo.ac.jp † Email: gucci"at"phys.titech.ac.jp ‡ Email: yokoyama"at"resceu.s.u-tokyo.ac.jp tion (NEC) is satisfied in the expanding phase, the Hubble parameter and the energy density of the universe increase backward in cosmic time. So, it is often claimed that, if one tries to discuss what happened before inflation and/or how inflation started, one needs to know the information of very high energy physics, and challenge the initial singularity problem [9] in terms of quantum gravity. But, this is not always the case.
Recently, it was recognized that, if an action includes higher derivative terms of a scalar field like the Galileon terms, the NEC can be violated without introducing ghost nor gradient instabilities. See, e.g., Ref. [10] for a recent review and Ref. [11] for a subtle issue of nonlinear instabilities. If the NEC is violated, the energy density can grow as time proceeds, contrary to the conventional wisdom. In the NEC violating theories, the universe can therefore start from the static zero-energy state described by the Minkowski spacetime from infinite past [12], and the universe starts expansion with the increase of the energy density.
Such a picture of the emergence of the universe was first proposed by Creminelli et al. [13] with the name Galilean Genesis. In their model, however, the hot big bang state was postulated to be realized after the effective field theory description breaks down as the energy density increases beyond its realm of validity. Therefore, the theory to describe the most important epoch of the early universe is lacking there.
Nevertheless, since their original idea is so interesting that a number of extension has been made in a wider class of scalar field theories [14][15][16][17][18] and various aspects of the Genesis scenario have been explored in the literature [19][20][21][22][23], such as avoidance of the superluminal propagation of perturbations and absence of primordial tensor perturbations. They have been unsuccessful, however, to realize transition from the Genesis phase to the hot big bang state within their model Lagrangians.
In this paper, we take a different approach, namely, to make use of the Galilean Genesis to explain the initial condition of the Universe before inflation and smoothly connect it to the inflationary phase, thereby solving the initial singularity problem [9] and the trans-Planckian problem [24] (see also [25]) in inflationary cosmology.
In fact, such an approach has also been put forward by Pirtskhalava et al. [26] recently. Their model Lagrangian, however, gives rise to gradient instability, which is postulated to be ameliorated only by higher order effects. Discussion on termination of inflation and reheating is also absent there.
In the present paper, we construct a concrete model free from any catastrophic instabilities and with subluminal velocities of primordial perturbations. In our setup the universe starts from the Minkowski spacetime from infinite past and is smoothly connected to the inflationary phase followed by the graceful exit. For this purpose, we provide a generic Lagrangian capable of describing the background and perturbation evolution in all the above phases instead of choosing the effective field theory approach because the latter cannot capture the evolution of the background and perturbations from pre-inflationary Genesis to the exit from inflation with the same single Lagrangian.
Although we start with asymptotically Minkowski space at the past infinity for aesthetic beauty, it has been shown that the Galilean Genesis solution is an attractor for a variety of initial conditions including those with a negative Hubble parameter and/or finite curvature, provided that the time derivative of the scalar field has the right sign [18]. The Horndeski theory [27] or the generalized Galileon [28], whose mutual equivalence was first shown in [29], is known to be the most general scalar-tensor tensor theory with the second-order field equations, and thereby avoid Ostrogradski instabilities in spite of having higher derivative terms in the action. The theory can be generalized to have second-order field equations only in a specific gauge while maintaining the number of propagating degrees of freedom. This possibility was realized recently by Gleyzes et al. [30] and was extended further by Gao [31]. The number of propagating degrees of freedom in these theories is indeed shown to be the same as that of the Horndeski theory [30][31][32][33][34]. In this paper, we use the subclass of Gao's framework as a concrete realization of the unified scenario starting from Galilean Genesis through inflation to the graceful exit.
This paper is organized as follows. In the next section, we give a framework of our model and derive the background equations of motion and the quadratic actions of cosmological perturbations. In Sec. III, a concrete Lagrangian is constructed to describe our scenario be-ginning from the Genesis phase through the inflationary one to the graceful exit, and such a background dynamics is presented explicitly. In Sec. IV, we discuss the stability during each phase based on the quadratic actions of cosmological perturbations. In Sec. V, a concrete realization of our scenario is given. The final section is devoted to our conclusions and discussion.

II. GENERAL FRAMEWORK
Let us start with describing the general framework to construct and study our explicit realization of the earlytime completion of inflation. We would like to consider theories composed of a metric g µν and a single scalar field φ, and hence it will be appropriate to work in the Horndeski theory. The Lagrangian of the Horndeski theory is of the form where X := −g µν ∂ µ φ∂ ν φ/2, R (4) is the four-dimensional Ricci scalar, and G (4) µν is the four-dimensional Einstein tensor. We have four arbitrary functions of φ and X in the Horndeski theory. This is the most general Lagrangian having second-order field equations. Nevertheless, it will turn out that this framework is insufficient for our purpose, and hence we have to go beyond the Horndeski theory.
One can generalize the Horndeski theory to possess higher order field equations while maintaining the number of propagating degrees of freedom [30]. The first step to do so is to perform an ADM decomposition by taking φ = const hypersurfaces as constant time hypersurfaces. In the ADM language, the metric is written as By definition φ is a function of only t, φ = φ(t), and X =φ 2 /2N 2 , where a dot denotes differentiation with respect to t, so any function of φ and X can be regarded as a function of t and the lapse function N , provided thatφ and N −1 never vanish. Then, the Horndeski Lagrangian (1) can be written in terms of the ADM variables as L = √ γN a L a with where K ij and R ij are the extrinsic and intrinsic curvature tensors on the constant time hypersurfaces, and A 4 , A 5 , B 4 , and B 5 are subject to the relations Variation of the above Lagrangian with respect to N gives a second-class constraint that eliminates only one degree of freedom, as opposed to general relativity. The key trick to generalize the Horndeski theory is to notice that this property remains the same even if one liberates A 4 and A 5 from the restriction imposed by Eq. (4) [30]. We thus arrive at the so called GLPV theory that is more general than Horndeski but has the same number of propagating degrees of freedom. One can move back to a covariant form of the Lagrangian by introducing the unit normal to the constant time hypersurfaces as n µ = −∂ µ φ/ √ 2X, writing the extrinsic curvature tensor in terms of n µ , and using the Gauss-Codazzi equations. Since there are six arbitrary functions of t and N in the ADM form, the resultant covariant Lagrangian has six arbitrary functions of φ and X.
The above idea has been pushed forward by Gao [31], who proposed a unified framework to study single scalartensor theories beyond Horndeski. One can write a general Lagrangian in the ADM form as where the coefficients d 0 , d 1 , ... are arbitrary functions of t and N . The Hamiltonian depends nonlinearly on N as in the GLPV theory, giving rise to a single scalar degree of freedom on top of the traceless and transverse gravitons [34].
In this paper, we will employ the Lagrangian L = √ γN a L a with where λ 1 , λ 2 , and λ 3 are constant parameters of the theory. This is a deformation of the GLPV Lagrangian and belongs to a subclass of Gao's framework. The generalization to this level is sufficient for the purpose of the present paper. The GLPV theory is recovered by taking Given the Lagrangian (6) in the ADM form, one can restore the scalar degree of freedom φ to write its covariant expression in the same way as in the GLPV theory. However, it will be more convenient for our purpose to use the explicitly time-dependent Lagrangian, because by doing so one can easily design the Lagrangian so as to admit the desired cosmological evolution.
Before specifying the suitable form of A 2 (t, N ), A 3 (t, N ), ... to construct our early universe model, let us derive the general equations governing the background and perturbation dynamics of cosmologies based on the Lagrangian (6). The ADM variables are given by where ζ is the curvature perturbation in the unitary gauge and h ij is the transverse and traceless tensor perturbation. A spatially flat background has been assumed and the spatial diffeomorphism invariance was used to write γ ij in the above form. In the following, the background value of the lapse function is denoted by N where there is no worry about confusion.

A. Background Equations
Substituting Eq. (7) to the Lagrangian (6), we obtain the background part of the Lagrangian as where η 4 := (3λ 1 − 1)/2, η 5 := (9λ 2 − 9λ 3 + 2)/2, and H :=ȧ/(N a). At the background level, λ 1 , λ 2 , and λ 3 just rescale A 4 and A 5 . In what follows we simply consider the case with η 4 > 0 ⇔ λ 1 > 1/3. Since we are considering a spatially flat universe, we have R ij = 0 at zeroth order, and hence B 4 and B 5 play no role in the background dynamics. Varying Eq. (8) with respect to N and a, we obtain, respectively, where a prime represents differentiation with respect to N . The background equations contain at most second derivatives of the scale factor and first derivatives of the Lapse function.

B. Cosmological Perturbations
The quadratic Lagrangian for the tensor perturbation is given by where The equation of motion contains at most second derivatives both in time and space. The tensor perturbation is stable provided that G T > 0 and F T > 0. The quadratic Lagrangian for the scalar perturbations is given by where the coefficients are defined as (19) and note the relation G T = G A − 3C. One has C = 0 in the Horndeski and GLPV theories, in which λ 1 = λ 2 = λ 3 = 1. Therefore, the last term in the Lagrangian (14) is the novel consequence of theories beyond GLPV.
From δL (2) S /δ(δn) = 0 and δL (2) S /δ(∂ 2 χ) = 0 we obtain Substituting Eqs. (20) and (21) into Eq. (14), we obtain the reduced Lagrangian for the curvature perturbation, where Thus, if C = 0, the equation of motion for ζ has the fourth derivative in space, giving the dispersion relation We require that G S > 0 in order to avoid ghost instabilities. However, we allow for a negative sound speed squared, c 2 s := F S /G S < 0, for a short period of time. In the absence of the k 4 term, a negative sound speed squared would cause a rapid growth of instabilities for large k modes. In this paper, we consider theories with C = 0, so that the curvature perturbation with large k can be stabilized by requiring that H S /G S > 0.

A. Construction of the Lagrangian
The Lagrangian we study in this paper is characterized by a single time-dependent function f (t) and four functions a 2 , a 3 , a 4 , a 5 of N : where α (> 0) is a constant parameter. We have introduced the mass scales M a (and the Planck mass M Pl ), so that f (t) and a a (N ) are dimensionless. The other two functions, B 4 and B 5 , are arbitrary at this stage because they have no impacts on the background dynamics. Note that f is not a dynamical variable. Specifying the functions f = f (t) and a a = a a (N ) amounts to defining a concrete theory. We design f (t) so as to implement the (generalized) Galilean Genesis followed by inflation and a graceful exit from the prolonged inflationary phase. Our choice is well before t = t 0 , and for t t 0 . As our time variable starts at t = −∞ with asymptotically Minkowski spacetime configuration, t is large and negative in the beginning, so we find f ≫ 1 in Eq. (31). As will be seen shortly, the initial stage described by Eq. (31) corresponds to the generalized Galilean Genesis, while the subsequent stage described by Eq. (32) to inflation. After a sufficiently long period of the inflationary stage, we assume that for t t end , where t end is the time at the end of inflation. With this the universe exits from inflation. In what follows we will investigate the background evolution of each stage.

B. Genesis Phase
Assuming that H ∼ |t| −(2α+1) in the first stage where f is given by Eq. (31), let us look for a consistent solution for large f . The background field equations read It can be seen from Eq. (34) that the lapse function N is a constant, N = N 0 , satisfying Then, H is consistently determined from Eq. (35), which can be written as is a constant. This leads to the generalized Galilean Genesis solution [18]: It is required thatp/η 4 < 0 to guarantee H > 0. We have thus arrived at the generalized Galilean Genesis solution starting from the Lagrangian written in the ADM form rather than in the covariant form. The original Galilean Genesis solution found in Ref. [13] corresponds to α = 1. In deriving the above solution, M 2 4 f −2α a 4 (⊂ A 4 ) and M 5 f a 5 (= A 5 ) are always subdominant due to the assumed scalings ∼ f −2α and ∼ f . Therefore, any choices of a 4 (N ) and a 5 (N ) will not spoil the above Galilean Genesis solution. As will be seen in the next section, those two terms are also irrelevant to the stability conditions during the Genesis phase.

C. Inflationary Phase
The Galilean Genesis phase will end at t ∼ t 0 since the function f is constant for t t 0 . In the subsequent phase we obtain the de Sitter solution, N = N inf = const and H = H inf = const, satisfying (Note that A a is now a function of N only and is independent of t.) A t-independent Lagrangian in the ADM form can be recast in a covariant Lagrangian with the shift symmetry, φ → φ + c. This implies that the above exact de Sitter solution corresponds to kinetically driven G-inflation. If one invokes a weak time-dependence in f , one obtains quasi-de Sitter inflation instead.

D. Graceful Exit
After the prolonged phase of inflation, f is given by Eq. (33). We assume that t is sufficiently large, so that f ≫ 1. Then, we have a consistent solution with N = N e = const and Thus, one can implement a graceful exit from inflation. It follows from Eq. (43) that (N e a 2 ) ′ < 0.
It can be shown using Eqs. (43) and (44) that, during this third stage, It is therefore necessary to impose m > 0 ⇔ a ′ 2 < 0. In the standard potential-driven inflation models [2] inflation is followed by coherent field oscillation of the inflaton scalar field which decays to radiation to reheat the universe. In the present approach the scalar field φ is used to specify constant time hypersurfaces, so thatφ may not vanish in order to preserve one-to-one correspondence between φ and the cosmic time t. Hence one must switch from the ADM language we used to construct the action to the conventional "φ language" at this point in order to apply the standard reheating mechanism, which is all right but looks like sewing a fox's skin to the lion's.
Here instead we consider another reheating mechanism which can take place without breaking the one-to-one correspondence between φ and t, namely, the gravitational reheating due to the change of geometry or the cosmic expansion law [35][36][37][38][39].
During the transition from the de Sitter inflation to a decelerated power-law expansion, conformally noninvariant particles are produced with the initial energy density where σ is a factor determined by the effective number of conformally noninvariant fields and the change of the geometry. For example, for m = 6 or 4, a single minimally coupled massless scalar field contributes to σ by respectively [39,40]. Here ∆t is the time required for the transition. In case it is nonminimally coupled with a coupling parameter ξ, a factor (1 − 6ξ) 2 is multiplied there.
In order for the radiation thus created to dominate the universe, the energy density of the scalar field must dissipate more rapidly, namely, then, the reheating temperature at the radiation domination is given by where g * is the effective number of relativistic degrees of freedom and we have assumed the universe would evolve in the same way as in the Einstein gravity after inflation. If long-lived massive particles are copiously produced at the gravitational particle production, the reheating temperature may be significantly higher then the above value. Furthermore, the decay of quasi-flat direction may produce a large amount of entropy to reheat the universe efficiently and create matter particles [41].

IV. PRIMORDIAL FLUCTUATIONS AND STABILITY
Having obtained the background evolution of our scenario, let us investigate the nature of primordial perturbations and stability, using the result of the generic analysis in Sec. II B.

A. Genesis Phase
During the Genesis phase, we have Obviously, the kinetic term of the tensor perturbations has the right sign, G T > 0. For large f , we see ΣC ≫ Θ 2 (as long as C = 0), and hence This implies that G S ≃ const, while H S ∼ (−t) 2(α+1) . The kinetic term of the curvature perturbation has the right sign if Thus, it is sufficient to impose (We are considering only the case with λ 1 > 1/3.) Another stability condition, H S > 0, is equivalent to requiring that Since F T depends on B 4 and B 5 and these two functions are irrelevant to the background dynamics, the condition F T > 0 can easily be satisfied without spoiling the Genesis background. Suppose for simplicity that where β (> 0) is a constant. Then, F T = G B = βM 2 Pl > 0. For the scalar perturbations we have This can also be made positive by an appropriate choice of a 3 (N ). It should be noted that if a 3 = 0 then we inevitably have F S < 0; the L 3 term is crucial for the stable violation of the NEC. Note also that, if we take sufficiently small β, the sound speed c s can be smaller than unity, which applies also to the other two phases discussed below. Let us move to discuss the nature of the primordial fluctuations in the Genesis phase. Since G T ∼ F T ∼ const, the tensor perturbations behave in the same way as in the Minkowski spacetime. Therefore, no large tensor modes are generated during the first stage of our scenario.
The behavior of the curvature perturbation turns out to be more nontrivial, as sketched in Fig. 1. Recalling that G S ∼ const, F S ∼ const, and H S ∼ (−t) 2(α+1) , the equation of motion for ζ in the Fourier space is of the form where y := −N 0 t > 0 and FIG. 1: Schematic diagram of the behavior of curvature perturbation in (y, a/k) plane with y decreasing toward the right. In the region below (above) the red broken curve, ω 2 is dominated by the term proportional to k 4 (k 2 ). Modes with k < k * experience the break down of the WKB approximation around the point crossing the blue solid curve beyond which ζ is frozen, while modes with k > k * do not.
with c s and k * being some constants. For sufficiently large y, we have ω 2 ≈ k 2α * k 4 y 2α+2 . One may define the time at which this approximation breaks down as y break := c 1/(α+1) s k −α/(α+1) * k −1/(α+1) , and for y ≪ y break we have ω 2 ≃ c 2 s k 2 . With some manipulation, it is found that where k * := c −(α+2)/α s k * . This implies that for the modes with k > k * the WKB approximation is always good in the Genesis phase, giving ζ k ∝ e icsky / √ c s k for y ≪ y break . Thus, the amplitude of those modes at late times in the Genesis phase is given by For the modes with k < k * , the WKB approximation breaks down at some time and then the curvature perturbation freezes. This "horizon crossing" occurs at y ∼ y freeze := k −α/(α+2) * k −2/(α+2) . It can be seen that y freeze > y break for k < k * , 1 which allows us to study the freezing process by using the solution to Eq. (59) with ω 2 ≈ k 2α * k 4 y 2α+2 . The exact solution in this case that matches the positive frequency WKB solution for y ≫ y freeze is given by where H (1) ν is the Hankel function of the first kind. The frozen amplitude can thus be evaluated by taking the limit y ≪ y freeze in the solution (64), leading to For y < y break , ω is dominated by the c s k term where the solution (64) is no longer exact. The frozen amplitude (65), however, is still valid even in this regime since the solution to Eq. (59) with the effective frequency (60) does not oscillate any more and remains constant. Hence, the expression of the power spectrum (65) is correct for the entire range of k <k * . To summarize, the power spectrum of the curvature perturbation generated during the Genesis phase is blue and hence is suppressed on large scales.

B. Inflationary Phase
In the (de Sitter) inflationary phase, G T , F T , G S , F S , and H S are time-independent. We require that all those coefficients are positive during inflation in order to avoid instabilities.
Since the quadratic action for the tensor perturbations is essentially the same as that of generalized G-inflation, the power spectrum of the primordial tensor perturbations is given by [29] 1 Note in passing that y break = y freeze = c 2/α s k −1 * for the k = k * mode. The Genesis phase could end sufficiently early so that * . If this is the case, we only need to care about the modes with k < k * .
The equation of motion for the canonically normalized variable u k := √ 2G S aζ k during inflation is of the form where with c 2 s = F S /G S and ǫ : being dimensionless constants. Here, we have introduced the conformal time τ (< 0) defined by adτ = N dt. The dispersion relations of this form have been studied in the context of inflation, e.g., in Refs. [42,43]. The positive frequency modes are given by [43] where W κ,m is the Whittaker function. Taking the limit τ → 0, the power spectrum of the curvature perturbation can be calculated as where Even in the presence of the k 4 term in the dispersion relation, the power spectrum is scale-invariant in the case of exact de Sitter inflation. Since we have F → 1 as x → ∞, we recover the result of generalized Ginflation [29] in the limit ǫ → 0. For x ≪ 1 we have F ≃ (4/π)|Γ(5/4)| 2 x −3/2 , so that one can take the limit c 2 s → 0 smoothly to get We have approximated the inflationary phase as exact de Sitter. If we consider a background slightly different from de Sitter by incorporating weak time dependence in f , we would be able to obtain a tilted spectrum of ζ.

C. Graceful Exit
After inflation, we have G T ≃ M 2 Pl , F T = βM 2 Pl , where to simplify the expression we introduced Recalling that we have been imposing λ 1 > 1, all of these coefficients are positive provided that ℓm > 2(λ 1 − 1). This condition can be written equivalently as

V. A CONCRETE EXAMPLE
Let us provide a concrete Lagrangian exhibiting the Genesis-de Sitter transition. The Lagrangian is characterized by where N 0 (> 0) and γ (> 0) are constants. We take a 4 = a 5 = 0, B 4 = M 2 Pl /2, and B 5 = 0. We also take λ 1 > 1 to guarantee the stability. This corresponds to the (λ 1 > 1 generalization of the) unitary gauge description of the Lagrangian considered in Ref. [13]. In the Genesis stage we have Since λ 1 > 1 and (N 0 a ′ 2 ) ′ = 2/N 2 0 > 0, we see that G S > 0 and H S > 0. We also see that and hence it is easy to satisfy F S > 0 during the Genesis phase by choosing the parameters appropriately. A numerical example of the Genesis-de Sitter transition is illustrated in Figs. 2 and 3. Our numerical calculation was performed as follows: we solve the evolution equations P = 0 and dE/dt = 0 with initial data (H, N ) satisfying E = 0, and confirm that the constraint E = 0 is satisfied at each time step. In the numerical calculation, the parameters are given by M Pl = M 2 = M 3 = 1, α = 1, λ 1 = 1 + 10 −3 , N 0 = 1, and γ = 10. The function f (t) is taken to be f =ḟ withḟ 0 = −10 −1 , f 1 = 10, and s = 2 × 10 −3 . The background evolution is shown in Fig. 2. The evolution of the sound speed squared, F S /G S , and the coefficient of k 4 in the dispersion relation is shown in Fig. 3. As pointed out in Ref. [26], c 2 s flips the sign at the transition. The sound speed squared is positive except in this finite period. During the Genesis and subsequent de Sitter phases we have G S > 0 and H S > 0, and therefore we may conclude that this model is stable.
Although we have thus obtained the stable example of the Genesis-de Sitter transition, the simple example (78) is not completely satisfactory if one would want successful gravitational reheating. Indeed, the condition (45) implies that x := (N e /N 0 ) 2 < 1, but m−4 = −2x/(1−x) < 0 for such x. This problem can be evaded easily by the following small deformation of a 2 : where ∆ is a parameter smaller than 1/5. The condition (45) now reads (1 − x)(x − 5∆ 2 ) > 0, i.e., 5∆ 2 < x < 1, while is positive for 5∆ 2 < x < ∆. The stability condition further restricts the allowed ranges of x and ∆. The necessary condition for stability is N e a ′ 2 /(N 2 e a ′ 2 ) ′ < 0 [see Eq. (77)].
To illustrate the final stage of inflation, let us take where the origin of time is shifted so that the end of inflation is given by t ∼ 0. In the numerical plots presented in Figs. 4 and 5, the parameters are given by s ′ = 10 −2 , v = 6, and ∆ = 0.05, while the other parameters are taken to be the same as the previous example of the Genesis-de Sitter transition. It is found that m ∼ 4.5 > 4. Again, we see that c 2 s < 0 in the finite period around the transition. However, G S and H S remain positive all through the inflation and subsequent stages.

VI. DISCUSSION AND CONCLUSION
In this paper, we have introduced a generic description of Galilean Genesis in terms of the ADM Lagrangian and constructed a concrete realization of inflation preceded by Galilean Genesis, i.e., the scenario in which the universe starts from Minkowski spacetime in the asymptotic past and is connected smoothly to the inflationary phase followed by the graceful exit. Our model utilizes the recent extension of the Horndeski theory, which has the same number of propagating degrees of freedom as the Horndeski theory and thus can avoid Ostrogradski instabilities. This approach allows us to cover the background and perturbation evolution in all the three phases with the same single Lagrangian, as opposed to the effective field theory approach. In our scenario, the sound speed squared during the transition from the Genesis phase to inflation becomes negative for a short period. However, thanks to the nonlinear dispersion relation arising from the fourth-order derivative term in the quadratic action, modes with higher momenta are stable and the growth rate of perturbations with smaller momenta is finite and under control. It should also be noted that the sound speed of the primordial perturbations can be smaller than unity by choosing the parameter of the model appropriately.
Although we have constructed our inflation model in order to resolve the initial singularity and possible trans-Planckian problems by incorporating Galilean Genesis phase before inflation, we could make use of our model to realize the original Galilean Genesis scenario, which is an alternative to inflationary cosmology, simply by taking vanishingly short period of inflation there. As discussed in the Appendix, the sound speed squared becomes negative at the transition also in this case, but the instabilities are relevant only for small k modes thanks to the k 4 term in the dispersion relation. Thus, the transition from the Genesis phase to the reheating stage is described in a healthy and controllable manner.
In fact, it would be fair to say that such a cosmology works quite well among the proposed alternatives to inflation, because, in contrast with the bouncing cosmology, in which all the would-be decaying modes in the expanding universe such as vector fluctuations and spatial anisotropy severely increase in an undesirable manner, the Genesis solution is an attractor and generation of nearly scale-invariant curvature perturbation is also possible with an appropriate choice of model parameters [18]. Since no first-order tensor perturbation is generated in this type of scenarios, detection of tensor perturbation with its amplitude larger than 10 −10 would be a smoking gun of inflation.