PHYSICAL REVIEW D 100, 123531 (2019) Birth of de Sitter universe from a time crystal universe † Daisuke Yoshida * and Jiro Soda Department of Physics, Kobe University, Kobe 657-8501, Japan (Received 30 September 2019; published 18 December 2019) We show that a simple subclass of Horndeski theory can describe a time crystal universe. The time crystal universe can be regarded as a baby universe nucleated from a flat space, which is mediated by an extension of Giddings-Strominger instanton in a 2-form theory dual to the Horndeski theory. Remarkably, when a cosmological constant is included, de Sitter universe can be created by tunneling from the time crystal universe. It gives rise to a past completion of an inflationary universe. DOI: 10.1103/PhysRevD.100.123531 I. INTRODUCTION Euler-Lagrange equations of motion contain up to second order derivatives. Actually, bouncing cosmology in the Inflation has succeeded in explaining current observa- Horndeski theory has been intensively investigated so far tions of the large scale structure of the Universe [1,2]. [23–29]. These analyses show the presence of gradient However, inflation has a past boundary [3], which could be instability [30,31] and finally no-go theorem for stable a true initial singularity or just a coordinate singularity bouncing cosmology in Horndeski theory was found for the depending on how much the Universe deviates from the spatially flat universe [32]. However, it was shown that the exact de Sitter [4], or depending on the topology of the no-go theorem does not hold when the spatial curvature is Universe [5]. It is also shown that the initial singularity included [33]. Therefore, we should consider Horndeski appears even in other scenarios [6,7]. The incompleteness theory in the presence of a spatial curvature in order to have of the inflationary universe strongly motivated us to explore a bouncing universe [34]. nonsingular scenarios in the very early Universe. The Recently, it was found that a subclass of Horndeski bouncing universe (see, e.g., [8] for a review) is an theory possesses a dual expression described by a 2-form interesting candidate of a nonsingular beginning of the gauge field [35], as in the case of a free massless scalar Universe. We can also consider a completely periodic universe as a possible beginning. A cyclic universe and field. In the case of the free massless field, the 2-form dual theory provides a topology changing tunneling process an ekpyrotic universe studied as an alternative to the 3 inflation [9–12] belong to this category. Since gravitational from a flat Euclidean space R to a direct sum space 3 ⊕ 3 theory has a symmetry of a time reparametrization, any S R which is mediated by the Giddings-Strominger 3 cyclic solution which is completely periodic in time is instanton [36]. The nucleated closed universe S is called called time crystal universe [13–18] by an analogy of a time baby universe. It is known that the baby universe contracts crystal in the condensed matter physics. after nucleation. Hence, it does not represent the realistic In this paper, we seek a cyclic universe as a past expanding universe. One possibility to circumvent this completion of an inflationary universe. Notice that, to conclusion would be to introduce an additional field which construct a bouncing solution, we need to violate the null provides a vacuum energy [37]. As an alternative possibil- energy condition. Since a scalar field minimally coupled ity, in this paper, we regard the contracting baby universe as with gravity in a flat universe satisfies the null energy a contracting phase of a bouncing/cyclic universe in condition, we need to consider more general situations. Horndeski theory. Since a spatial curvature is naturally It is Horndeski theory [19–21](see [22] for a recent review) introduced in this set up, there is a chance to realize a that is the general class of scalar-tensor theory whose bouncing universe. Indeed, we find a time crystal universe as a baby universe nucleated from a flat space, which is * mediated by an extension of Giddings-Strominger instan- [email protected] † [email protected] ton in a 2-form theory dual to the Horndeski theory. Interestingly, once a cosmological constant is introduced, Published by the American Physical Society under the terms of it turns out that de Sitter universe can be created from time the Creative Commons Attribution 4.0 International license. crystal. Thus, we have a completion of inflationary sce- Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, nario, namely, the past boundary of an inflationary universe and DOI. Funded by SCOAP3. is time crystal. 2470-0010=2019=100(12)=123531(11) 123531-1 Published by the American Physical Society DAISUKE YOSHIDA and JIRO SODA PHYS. REV. D 100, 123531 (2019) Z pffiffiffiffiffiffi M2 1 The paper is organized as follows. In the next section, we ¼ 4 − pl ð −2ΛÞ− Gμνρ αβγ ð Þ review the 2-form theory dual to a subclass of the shift S d x g 2 R 12 αβγHμνρH ; 4 symmetric Horndeski theory [35]. Then in Sec. III,we construct a time crystal universe in the 2-form dual of where H ¼ dB is the field strength of the 2-form gauge Horndeski theory. In Sec. IV, we solve Euclidean equation field Bμν. The components of H are given by of motion in 2-form theory without a cosmological constant and construct an extension of Giddings-Strominger instan- Hμνρ ¼ 3∂½μBνρ ¼ ∂μBνρ þ ∂νBρμ þ ∂ρBμν: ð5Þ ton. This instanton describes a nucleation of a time crystal μνρ universe from a flat space. Then, in Sec. V, we extend this The tensor G αβγ is defined by analysis in the presence of a cosmological constant and find Gμ Gν Gρ an instanton solution which mediates a tunneling process μνρ α β γ G αβγ ≔ ; ð6Þ from a time crystal universe to an inflationary universe. G· det · Thus, a past completion of an inflationary scenario is G· Gμ obtained. The final section is devoted to summary and where det · is the determinant of ν. The duality relation discussion. There we comment on the stability issue. is no longer given by the simple Hodge dual. It now depends on the curvature of spacetime as II. 2-FORM DUAL OF HORNDESKI THEORY ν 1 νρσ Gμ ∂νϕ ¼ ϵμ Hνρσ: ð Þ In this section, we review a 2-form gauge theory dual to a 3! 7 subclass of Horndeski theory [35]. We refer the reader to the original paper [35] for a detailed derivation. In the next section, we study the above action and obtain We focus on a shift symmetric scalar field coupling to an extension of the Giddings-Strominger instanton, which gravity through the Einstein tensor Gμν as, is an instanton solution in free 2-form theory with gravity. Z pffiffiffiffiffiffi M2 III. TIME CRYSTAL UNIVERSE WITH ϕ ¼ 4 − pl ð − 2ΛÞ S d x g 2 R 2-FORM CHARGE α β In this section, we construct a cosmological solution of − μν∂ ϕ∂ ϕ − μν∂ ϕ∂ ϕ ð Þ 2 g μ ν 2 G μ ν ; 1 the 2-form theory (4). Note that, thanks to the duality, following results can be obtained even if one starts with the where α and β are constants. We assume α > 0 to ensure scalar action (1). The essential difference will appear when that the kinetic term of the scalar field ϕ has a correct sign at we discuss quantum effects in the next section. the low energy. Though the Einstein tensor includes the We consider homogeneous and isotropic ansatz without second derivatives of the metric, equations of motion specifying the spatial curvature, include up to the second derivatives. Hence, this interaction μ ν ¼ 2½− ð Þ2 2 þ ð Þ2Ω i j ð Þ does not have the Ostrogradsky’s ghost. Actually, this is a gμνdx dx l N t dt a t ijdx dx ; 8 subclass of Horndeski theory with G2 ¼ α=2, G4 ¼ Ω 2 2 þ β ¼ ¼ 0 where is given by Mpl= X and G3 G5 , where X stands for μν g ∂μϕ∂νϕ. This class of theory was, for example, studied i j 2 2 2 2 2 Ωijdx dx ¼ dχ þ fkðχÞ ðdθ þ sin θdϕ Þ; to construct hairy black hole solutions [38]. For conven- 8 μν χ ðk ¼þ1Þ ience, we introduce the effective metric G by < sin fkðχÞ¼ χ ðk ¼ 0Þ : ð9Þ Gμν ¼ αgμν þ βGμν: ð2Þ : sinh χ ðk ¼ −1Þ Then, the action can be written as Z Here k represents the sign of the spatial curvature. We have pffiffiffiffiffiffi M2 1 introduced a free parameter l with a mass dimension −1 so ϕ ¼ 4 − pl ð − 2ΛÞ − Gμν∂ ϕ∂ ϕ ð Þ S d x g 2 R 2 μ ν : 3 that the coordinates ðt; χ; θ; ϕÞ and the dynamical variables NðtÞ and aðtÞ are dimensionless. We will also use the As is well known, a free massless scalar field is (dimensionless) proper time τ, that is defined by dτ ¼ Ndt. equivalent to a free 2-form gauge field through the duality, We assume homogeneous and isotropic configuration of dϕ ¼dB, where B is a 2-form field and à represents the the field strength H; pffiffiffiffi Hodge dual. In Ref. [35], it was shown that the similar H ¼ h Ωdχ ∧ dθ ∧ dϕ; ð10Þ duality holds even if the derivative coupling through the Einstein tensor (1) is included. The resultant action is where h represents the magnetic flux density of the 2-from given by gauge field.