Birth of De Sitter Universe from a Time Crystal Universe

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 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. The symmetry allows h to depend on time but

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V V 1.0 1.0

0.5 0.5 V1 (k = 1)

V1(k = 1) V2 (k = 1) a V (k = 0) V (k = 1) 0.5 1.0 1.5 2.0 2.5 3.0 1 a 3 1 2 3 4 5 V1(k = –1) V1 (k = 0) –0.5 V1 (k = –1) –0.5

–1.0 –1.0

1 FIG. 1. Potential for positive γ (left: γ ¼ 4, right: γ ¼ 4) with λ ¼ 0: The blue plot in the left figure provides an oscillating solution between amin ∼ 0.5 and amax ∼ 1.08. Vn which are not written in the plot are complex value. Z 2 2 it is actually forbidden from the Bianchi identity dH ¼ 0. ∂τ a ∂τa S ¼ 3l2M2 V dtNa3 þ One can check that our ansatz (10) satisfies the equation of pl a a motion as well: k λ 1 1 þ − − ; ð15Þ μνρ αβγ 2 3 6 1 − γ2ðð∂ Þ2 þ 2Þ ∇μðG αβγH Þ¼0: ð11Þ a a τa=a k=a

Now, we fix the free parameter l by V whereR isp theffiffiffiffi comoving volume of constant time slices, 3 2 1=4 V ≔ d x Ω. Taking the variation of the action with ≔ h ð Þ l 6α 2 : 12 respect to N, we obtain the modified Friedmann equation, Mpl β 2α λ − 2 þ γð3ð∂ Þ2 þ Þ Then we introduce a dimensionless ratio of to l , say ð∂ Þ2 þ − 2 þ a τa k ¼ 0 ð Þ γ τa k a 3 2 2 : 16 ,by 3 ða − aγðð∂τaÞ þ kÞÞ rffiffiffi 3 β 6 3β Mpl 2 γ ≔ ¼ ; ð13Þ By solving it for ð∂τaÞ , we derive the convenient form α l2 α jhj

λ 2 and a dimensionless cosmological constant by ð∂τaÞ ¼ −Vða; γ;k;λÞ; ð17Þ λ ≔ l2Λ: ð14Þ where the effective potential V has 3 branches correspond- Now, we obtain the minisuperspace action ing to 3 roots of the cubic equation;

1=3 6 2 6 þ λγ 2 − 2πð −1Þ Ξ 2πð −1Þ −81γ þ a ð−3 þ γλÞ V ða; γ;k;λÞ¼k − a þ e i 3 n þ ei 3 n ; ðn ¼ 1; 2; 3Þð18Þ n 9γ 9aγ2 9aΞ1=3 with

243 27 Ξð ; γ λÞ γ3 ¼ −729γ2 − 9ð−3 þ γλÞ3 þ γ 3ð3 þ γλÞþ ½γ3ð2916γ2 þ 2916γ3 2 2 þ 8γ 10ð−3 þ γλÞ3 a ;k; = ka a 2 a 2γ k a ka − 972γ2ka4ð3 þ γλÞ − 8a12ð−3γλÞ3 − 27γa6ðγ2λ2 − 42γλ þ 9ÞÞ1=2: ð19Þ

Here we define a branch cut of the power function as with the potential Vn and a total energy E ¼ 0. The typical n n inArgz z ≔ jzj e ; Argz ∈ ð−π; π. potentials Vn for λ ¼ 0 are plotted in Fig. 1 (positive γ) and The behavior of the solution can be understood by the Fig. 2 (negative γ). The most interesting solution is that of analogy of the dynamics of a point particle: Eq. (17) is k ¼ 1 with small positive γ (the blue plot in the left of ¼ nothing but the energy conservation law of a point particle Fig. 1), where the potential has 2 turning points a amin

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V V 10 4 V1 (k = 1) V1 (k = 1) 8 V2 (k = 1) V2 (k = 1) 2 6 V3 (k = 1) V3 (k = 1) V (k = 0) V (k = 0) 4 1 1 V2 (k = 0) a V2 (k = 0) 2 0.5 1.0 1.5 2.0 2.5 3.0 V3 (k = 0) V3 (k = 0) a V (k = –1) V (k = –1) 0.5 1.0 1.5 2.0 2.5 3.0 1 1 V (k = –1) –2 V (k = –1) –2 2 2 V3 (k = –1) V3 (k = –1) –4 –4

1 FIG. 2. Potential for negative γ (left: γ ¼ − 4, right: γ ¼ −4) with λ ¼ 0: No potential has an oscillating solution. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and a , which are 0.5 and 1.08 in Fig. 1 respectively. The pffiffiffiffiffiffiffiffiffiffiffiffiffi max pffiffiffi γ þ 4 þ γ2 scale factor will oscillate between amin and amax. Thus the a ≔ γ;a≔ ; ð20Þ solution is expected to express a time crystal universe, min max 2 where scale factor is exactly periodic. We will investigate this solution analytically in the next subsection. Note that which corresponds to the minimum and maximal size of the any potential other than k ¼ 1 with small positive γ has at oscillating universe. Since a time coordinate has not fixed most one terming point and does not have a time crystal yet, we can choose it by fixing a functional form of aðtÞ solution. freely. Our choice of the time coordinate t is as follows: Once the cosmological constant λ is turned on, the potential changes as in Fig. 3. For a negative λ (blue), there ð þ Þ − ð − Þ 2 λ ¼ 0 ð Þ ≔ amin amax amin amax cos t ð Þ is no essential difference from the case. For a large a t 2 : 21 positive λ (orange), the time crystal solution no longer exists and universe enters the λ dominant eta. For a small positive λ (red), the potential has three zero points, a ¼ Then the lapse function associated with this time coordinate ∼ 0 5 is given by plugging (21) into (17): amin;amax and aΛ, a . , 1.12 and 2.4 in Fig. 3. The time crystal universe, which oscillates between amin and amax, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi still exist. In addition, there is another solution in a>aΛ. 22 ð Þ¼ð − Þ sin t ð Þ Since the cosmological constant dominates over any other N t amax amin : 22 −V1ðaðtÞÞ term as a grows, this solution asymptotically approaches a de Sitter spacetime. In Sec. V, we will see that the time crystal universe can make a transition quantum mechan- Note that the analytic expression of V1 is given by Eq. (18). ically into the inflationary universe. Equations (21) and (22) provide an analytic solution, which is plotted in Fig. 4: A. Analytic solution Note that if the choice of time coordinate (21) were ill defined at some point, a singular behavior appears in NðtÞ, We would like to construct the analytic solution for the which tell us a presence of coordinate singularity. For case k ¼ 1, λ ¼ 0, and a small positive γ. It is useful to find example, if one choose a time coordinate t by a ¼ t,a the analytic formulas for amin and amax by solving Eq. (17) 2 ∂ ¼ 0 positive N can be obtained only for amin

V 1.0 1.2

0.5 1.0 V1 ( =–1/2)

V1 ( =0) 0.8 a 0.5 1.0 1.5 2.0 2.5 3.0 V1 ( =1/2) N V1 ( =2) 0.6 –0.5 a 0.4

–1.0 0.2

1 FIG. 3. Potential with cosmological constant: γ ¼ 4: A small t 3 5 0 2 3 cosmological constant (red plot) provides a new branch of 2 2 2 solution a>aΛ ∼ 2.4 in addition to the oscillating solution ∈ ð ∼ 0 5 ∼ 1 12Þ a amin . ;amax . . FIG. 4. Plot of the lapse function N and the scale factor a.

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a express our result by the proper time. The proper time τ can

1.0 be obtained by integrating N, Z t 0.8 τðtÞ¼ dtN: ð23Þ 0

0.6 The scale factor can be written as a function of this proper time, which is plotted in Fig. 5. Here the period of scale 0.4 factor in terms of the proper time is τðt ¼ πÞ ∼ 3.05 when γ ¼ 1. The proper time τðπÞ depends on the value of γ as 0.2 4 shown in Fig. 6. As another coordinate independent expression of our 2 4 6 8 result, it is useful to embed the 4-dimensional time crystal universe to the 5-dimensional Minkowski spacetime, FIG. 5. The scale factor a as a function of the proper time τ: scale factor a is a periodic function of τ. 2 2 2 2 i j l ½−dz þ da þ a Ωijdx dx ; ð24Þ ¼ ð Þ ¼ ð Þ (t= ) by the embedding function z z t and a a t . Here, 3.5 aðtÞ is the scale factor of the time crystal universe while the function z ¼ zðtÞ can be determined by imposing that the 3.0 pull back of 5-dimensional Minkowski metric coincides

2.5 with our time crystal universe, that is,

2 2 2 2 2 i j 2.0 l ½−ðð∂tzðtÞÞ − ð∂taÞ Þdt þ aðtÞ Ωijdx dx 2 2 2 2 i j 1.5 ¼ l ½−N dt þ a ðtÞΩijdx dx : ð25Þ

1.0 Then it can be integrated as

0.5 Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t 2 2 zðtÞ¼ dt N þð∂taÞ ; ð26Þ 0.0 0 0.0 0.5 1.0 1.5 2.0 2.5

FIG. 6. γ dependence of the period of oscillation measured by where the initial condition at t ¼ 0 is set to the origin of z. the proper time τ. Actually, we can visualize this embedding as a 2-dimensional surface in the 3-dimensional Minkowski spacetime π The numerical plot in Fig. 4 shows that our choice of a time by considering a subspace χ ¼ θ ¼ 2, where χ and θ are the coordinate (21) has no coordinate singularity. polar coordinates given by Eq. (9) with k ¼þ1. The Since the expression of our solution in Fig. 4 highly embedding can be seen in Fig. 7. Embedding the solution depends on our choice of a time coordinate, it is useful to into the flat space will help us to understand the analytic

FIG. 7. 2-dimensional time crystal universe (setting χ ¼ θ ¼ π=2) embedded in 3-dimensional Minkowski spacetime with a time coordinate z.

123531-5 DAISUKE YOSHIDA and JIRO SODA PHYS. REV. D 100, 123531 (2019) continuation of an Euclidean solution to a Lorentzian 0 ¼ δ S ZB E solution as we will see in the following sections. pffiffiffiffiffi 1 ¼ 3 − ∇ ðG μνρ αβγÞδ dtEd x gE 2 μ E αβγH Bνρ IV. NUCLEATION OF TIME CRYSTAL UNIVERSE 1 FROM FLAT SPACE þ ∇ ðG μνρ αβγδ Þ ð Þ 2 μ E αβγH Bνρ : 32 Since the duality is an on-shell relation, quantum effects such as the tunneling effect are expected to show Note that the second term automatically vanishes under our differences. The Euclidean solution with the free 2-form ansatz (29) and (30). That ensures that we do not need to field is known as a Giddings-Strominger instanton [36], introduce an additional boundary term with respect to the which provides a topology changing tunneling process 2-form field even for a boundary condition δBμν ≠ 0. 3 3 from a flat space R to a sum of a flat space R and a closed Now the Euclidean action for the minisuperspace can be Friedmann-Lemaître-Robertson-Walker (FLRW) universe obtained as 3 Z S called a baby universe. In the following, we will 2 2 ∂τ a ∂τ a investigate how the Giddings-Strominger instanton is S ¼ 3l2M2 V dt Na3 E þ E E pl E a a modified in the 2-form dual of Horndeski theory. To construct an instanton solution, we first need to λ 1 1 − k þ þ ð Þ t → −it 2 6 2 2 2 : 33 perform the Wick rotation, E, to obtain an a 3 a 1 þ γ ðð∂τa=aÞ − k=a Þ Euclidean action: Note that our new interaction term does not include the second derivative of the scale factor. That means we only S ¼ −iSj →− E t itE need the Gibbons-Hawking term of the Einstein-Hilbert Z 2 3 pffiffiffiffiffi Mpl term for gravitational variation in minisuper space. By ¼ dtEd x gE − ðRE − 2ΛÞ 2 taking the variation of the Euclidean action SE with respect 1 to N, we can deduce the modified Euclidean Friedmann þ G μνρ;αβγ ð Þ equation as 12 E HμνρHαβγ : 27 2 2 λ −a þγð−3ð∂τ aÞ þkÞ 0 ¼ −ð∂ Þ2 þ − 2 þ E ð Þ E E E τE a k a 3 2 2 : 34 Here gμν is the Euclidean metric, R and Gμν are the Ricci 3 ða −γað−ð∂τ aÞ þkÞÞ E E scalar and the Einstein tensor with respect to gμν. GE is the Euclidean version of (6), which is defined by From this expression, we see that the Euclidean Friedmann equation can be obtained from the Lorentzian Friedmann 2 2 μα νβ ργ ð∂τaÞ → −ð∂τ aÞ G G G μν equation simply by replacing E . Thus by G μνρ;αβγ ≔ E E E ; G μν ¼ αg þβG μν: ð28Þ E G · E E E defining the potential for the scale factor by det E· 2 E ð∂τ aÞ ¼ −V ða; γ; λ;kÞ;n¼ 1; 2; 3 ð35Þ Let us consider the Euclidean version of the FLRW E n ansatz, we find that the Euclidean potential is nothing but the negative signed Lorentzian potential, E μ ν ¼ 2½ ð Þ2 2 þ ð Þ2Ω i j ð Þ gμνdx dx l N tE dtE a tE ijdx dx ; 29 V with a constant magnetic flux of 2-form field, 1.0

pffiffiffiffi 0.5 E H ¼ h Ωdχ ∧ dθ ∧ dϕ: ð30Þ V1 E 2 V1 +O( ) a 0.5 1.0 1.5 2.0 2.5 3.0 As similar to the Lorentzian case, H satisfies the field V1 equation –0.5

μνρ αβγ –1.0 ∇μðGE αβγH Þ¼0; ð31Þ FIG. 8. Euclidean potential VE in 2-form theory without ¼ 0 1 as well as the Bianchi identity dH . Here we again set a cosmological constant (γ ¼ 4, k ¼ 1, λ ¼ 0): The gray plot is ∈ ð ∼ 0 5 length scale l by Eq. (12). The above field equation was the Lorentzian potential for time crystal a amin . ; ∼ 1 08Þ derived by taking a variation of the Euclidean action with amax . . The red, dashed plot is a Taylor expansion of E respect to Bμν as V1 given by (37).

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E Vn ða; γ; λ;kÞ¼−Vnða; γ; λ;kÞ: ð36Þ As seen from Fig. 8, in the a>amax region, the potential can be approximately expressed by the Taylor expansion with respect to γ, We note that this is because the 2-from field contribute to the Friedmann equation through a gradient energy, not a 1 1 1 VEða;γ;λ ¼ 0;k¼ 1Þ¼−1þ − −2þ γ þOðγ2Þ: kinetic energy. In the case of scalar theory, the contribution 1 a4 a6 a4 of the scalar field to the Friedmann equation is from the ð37Þ kinetic term and the last term in Eq. (34) receives an additional minus sign. As in the Lorentzian case, we can choose a time coordinate λ ¼ 0 The Euclidean potential for case is plotted in by fixing the functional form of aðt Þ. Our choice of the ∈ ð0 Þ E Fig. 8. There are 2 possible solutions, a ;amin and time coordinate is the following ∈ ð ∞Þ a amax; , where amin and amax are minimum and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi maximum size of the Lorentzian time crystal universe ð Þ ≔ ð Þ a tE amax cosh tE: 38 defined by (20). We are interested in the outer solution ∈ ð ∞Þ a amax; because it corresponds to the Giddings- Then the Friedmann equation provides the expression Strominger instanton. for N as

1 1 2 1 ð Þ2 ¼ þ − þ 2 γ þ Oðγ2Þ ð Þ N tE 4 cosh tE 8 cosh tE 1 þ 2 : 39 cosh tE cosh tE

Thus we obtain the solution, 1 γ 2 2 1 E μ ν ¼ 2 ð Þ þ 1 − þ 2 þ 1 þ γ Ω i j þ Oðγ2Þ ð Þ gμνdx dx l cosh tE 4 8 ð Þð1 þ Þ 3 dtE 2 ijdx dx : 40 cosh tE cosh tE cosh tE

Note that the leading term (γ ¼ 0) is nothing but the Giddings-Strominger solution. In the asymptotic region tE → ∞, our metric reduces that of the flat space μ ν 2 γ 2 1 2 2 2 2 E → 1 þ þ Oðγ Þ tE þ tE Ω i j ¼ ½ τ þ τ Ω i j ð Þ gμνdx dx l 2 4 e dtE e ijdx dx l d E E ijdx dx ; 41 where the proper time τ is given as Concretely, by embedding this space to a higher dimen- E 1=2 sional Euclidean space as Fig. 9, one can visually under- γ 2 2 τ ¼ 1 þ þ Oðγ Þ tE= ð Þ E 2 e : 42 stand the structure. Here embedding function is given as in the Lorentzian case, except for the sign of 5-dimensional The metric (40) represents a wormhole space which metric: connects two distinct asymptotic flat spaces tE → ∞.

2ð 2 þ 2 þ 2Ω i jÞ → 2ðð 0ð Þ2 þ 0ð Þ2Þ 2 þ ð Þ2Ω i jÞ¼ 2ð 2 2 þ ð Þ2Ω i jÞ ð Þ l dz da a ijdx dx l z tE a tE dtE a tE ijdx dx l N dtE a tE ijdx dx ; 43 where the embedding function can be obtained as Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t E 2 2 zðtEÞ¼ dtE N − ð∂t aÞ : ð44Þ 0 E

Based on the standard prescription of Wentzel-Kramers-Brillouin analysis, the tunneling probability from the flat space to the time crystal universe can be estimated by the on-shell value of Euclidean action. We find that the probability is finite: Z ∞ 1 γ 1 6 4 ¼ 6 2 2 π2 þ − þ − þ Oðγ2Þ SE l Mpl dtE 2 t 2 4 −∞ cosh t 4 cosh ð EÞ cosh t cosh t E 2 E E 2 ¼ 6l2M2 π3 1 þ γ þ Oðγ2Þ ; ð Þ pl 3π 45

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V 1.0

0.5 E V1 E 2 V1 +O( ) a 0.5 1.0 1.5 2.0 2.5 3.0 V

–0.5

–1.0 FIG. 9. 2-dimensional wormhole space embedded in 3-dimensional Euclid space: Each circles corresponds to tE constant FIG. 11. Euclidean potential VE in 2-form theory with a small curve. There are two asymptotic flat region (upper and lower 1 1 cosmological constant (γ ¼ 4, k ¼ 1, λ ¼ 2): The gray plot is the planes). Lorentzian potential and the red, dashed one is a Taylor expansion of Euclidean potential given by (46). Here the points E ¼ 0 ¼ 0 5 ¼ 1 12 ¼ 2 41 of V are given as amin . , amax ; , aΛ . .

A potential with a positive small λ is shown in Fig. 11. The tunneling from a time crystal universe to an inflationary one will be described a bounce solution between amax and aΛ. As seen from Fig. 11, the Taylor expansion of the potential is good approximation and that is given by 1 1 ð ; γ λ ¼ 1Þ¼−1 þ þ 2λ VE a ; ;k 4 3 a a 1 1 1 − −2 þ þ λa2 γ þ Oðγ2Þ: a6 a4 3 ð46Þ

FIG. 10. Analytic continuation of a Lorentzian time crystal As in the analysis of λ ¼ 0 case, one can freely choose a 0 universe (orange) from a wormhole space (blue): Here z> is time coordinate tE by fixing a functional form of aðtEÞ. Minkowski space and z<0 is Euclid space. The two sides of However, we can not introduce the global time coordinate ≔ ϕ → ∞ asymptotic flat regions (e.g., T a cos ) can also be analytically, simply because it is difficult to derive the connected to the Lorentzian Minkowski spacetimes by the analytical continuation based on Cartesian coordinates T → iT. analytic expression for the points amax and aΛ. Instead, we will introduce a time coordinate with coordinate singularity. and hence our time crystal universe can be nucleated from The simplest choice is the flat space. This scenario is visualized as in Fig. 10. ð Þ¼ ∈ ð Þ ð Þ Since two sides of the asymptotically flat region can be a tE tE tE amax;aΛ ; 47 connected to Minkowski spacetimes, this instanton describes a tunneling process from R3 to R3 ⊕ S3, where S3 is and the lapse function N can be obtained from the our time crystal universe. The tunneling probability of Friedmann equation as Giddings-Strominger is recovered by setting γ ¼ 0. Since 1 4 the correction γ is positive, nucleation probability of the 2 t Nðt Þ ¼ ¼ − E E − ð Þ 1 − 4 þðλ 3Þ 6 time crystal universe is slightly smaller than standard V tE tE = tE Giddings-Strominger’s baby universe. 4 6 1 − 2t þðλ=3Þt − E E γ þ Oðγ2Þ: ð48Þ t2 ð1 − t2 þðλ=3Þt6 Þ2 V. TUNNELING FROM THE TIME CRYSTAL TO E E E DE SITTER UNIVERSE ¼ There are coordinate singularities at a amax and aΛ In this section, we show, once the cosmological constant where NðtEÞ → ∞ (VðtEÞ¼0) and our time coordinate is tuned on, a time crystal universe will transit to an spans only a finite part of the whole space. By integrating inflationary universe by tunneling. N, we can obtain the proper time τ,

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a 2.5

2.0

1.5

1.0

0.5

E –3 –2 –1 1 2 3

FIG. 12. The scale factor a as a function of the proper time τE: Here only a period is plotted. Z t τð Þ¼ E ð Þ ∈ ð Þ ð Þ tE N s ds; tE amax;aΛ ; 49 aΛ FIG. 13. The instanton solution embedded in the flat Euclidean where we set τ ¼ 0 at the bounce point a ¼ aΛ. space. Numerically scalar factor can be plotted as a function of the proper time τE as the solid curve in Fig. 12. Similarly, the solution for tE >aΛ can be obtained by assuming instanton solution mediates a time crystal universe and de Sitter universe. Note that our result implies that our time ð Þ¼2 − ∈ ð 2 − Þ ð Þ a tE aΛ tE;tE aΛ; aΛ amax : 50 crystal universe is a false vacuum because it decays into de Sitter universe, which has a time translation symmetry. By repeating same procedure, we can plot the scale factor a τ ð 2 − Þ as a function of E for aΛ; aΛ amax , which is the dashed curve in Fig. 12. The embedding to the 5 dimensional Euclidean space can be obtained as in the λ ¼ 0 case. Now the embedding function z is given by Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t E 2 zðtEÞ¼ NðtEÞ − 1: ð51Þ aΛ

The embedding of the instanton solution is plotted in Fig. 13. Finally we need to confirm that the transition probability is finite. Since we know the exact solution (47) and (48),it is easy to estimate the value of the on-shell Euclidean action SE at least numerically. The results of numerical calcu- lations in the parameter region γ ∈ ð0.2; 0.8Þ and λ ∈ ð0.1; 0.8Þ can be fitted as 4.0 S ∼ 6π2l2M2 − þ 3 4 þ 1 1γ : ð Þ E pl λ . . 52

1 For example, we obtain SE ¼ −4.4 for the parameter γ ¼ 4 λ ¼ 1 and 2. Since the value of SE is finite, we can conclude FIG. 14. Tunneling process from time a crystal universe to de that our time crystal universe can make a transition into an Sitter universe mediated by our instanton solution: The orange inflationary universe through tunneling. The whole history surfaces are embedded in Minkowski spacetime as well as the of the tunneling process is described in Fig. 14. There our blue one is in Euclid space.

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VI. SUMMARY AND DISCUSSION and this leads to the curvature singularity at τ ¼ 0 because R ¼ 12=τ2 þ Oðτ0Þ. Though this does not lead to the We found that a 2-form gauge theory (4), as well as a E divergence of the Einstein Hilbert term because dual Horndeski theory (1), has a solution describing a time 3 ∝−τ crystal universe, which is exactly periodic in time as shown Na RE , the contribution from the 2-form kinetic term in Fig. 5. By considering the semiclassical effect of becomes negative infinity; quantum gravity, we found that (i) the time crystal universe pffiffiffi 1 pffiffiffiffiffi 9 3 2 can be nucleated from the flat space when Λ ¼ 0 and μνρ αβγ lMpl g G αβγHμνρH ¼ − þ OðτÞ → −∞: (ii) the time crystal universe decays into a de Sitter universe 12 E E 2γτ Λ 0 when > . The history of the transitions are visualized in ð56Þ Fig. 10 and Fig. 14. The last result suggest a new picture that the inflationary universe is created from a time crystal Thus if we assume that the tunneling probability is given by by tunneling. Thus, our finding gives rise to a past −S e E it diverges and dominates over the other processes completion of inflationary scenario. such as creation of the time crystal universe. However it Though our Euclidean solution provides a new interest- might be a matter of the prescription of quantum gravity. ing scenarios of the early Universe, there are problems For example, it is well known that the sign in front of SE is which need to be solved. The first problem would be the flipped depending on the boundary condition of the wave stability of the time crystal universe. Not only the time function of the universe (See, e.g., Ref. [39] for a review of crystal universe but any bouncing universe in Horndeski quantum cosmology). Apart from this, we can simply say theory tends to suffer from the stability issue [32]. In case that solution near a ¼ 0 is unreliable because a validity of of the universe with a spatial curvature, the stability is effective field theory of our action should be broken near studied [33] and found that the tensor perturbation is stable the curvature singularity. if and only if the following inequalities are satisfied: A possibility of the presence of the time crystal universe F ≔ 2 − β 0 G ≔ 2 þ β 0 ð Þ before the inflationary phase provides interesting questions. T Mpl X> ; T Mpl X> ; 53 For example, how can we know the signal of time crystal before tunneling process? We expect that the signal should F G where positivity of T and T ensures the absence of be encoded by cosmological perturbations, if the problem gradient and ghost instability respectively. Thus the abso- of ghost instability is resolved. As a virtue of time crystal j j lute value of the kinetic energy of scalar field X must be universe fields on the time crystal universe must be −∞ finite. However one can check that X goes to as the gradually amplified because of Floquet’s theorem, which scale factor a approaches to its minimum amin. Thus near is the time periodic version of Bloch’s theorem of usual the lower turning point amin, tensor modes become ghosts. crystals. Thus the cosmological perturbations are also Since our action includes a negative mass dimension expected to be amplified as the universe oscillates. In operator, it is natural to regard our theory as a low energy our scenario, the time crystal universe enters inflationary Λ effective theory with a cut off scale cut

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