Abstract
By including appropriate Riemanncubic invariants, we find that the dynamics of classical time crystals can be straightforwardly realized in Einstein gravity on the FLRW metric. The time reflection symmetry is spontaneously broken in the two vacua with the same scale factor a, but opposite \({\dot{a}}\). The tunneling from one vacuum to the other provides a robust mechanism for bounce universes; it always occurs for systems with positive energy density. For a suitable matter energymomentum tensor we also construct cyclic universes. Cosmological solutions that resemble the classical time crystals can be constructed in massive gravity.
Introduction
Recently, the fascinating concept of time crystal was proposed [1, 2], and subsequently realized in experiments [3,4,5]. A time crystal refers to a ground state that breaks the timetranslational invariance, such that it is periodic in both space and time. The subject has attracted considerable attention, e.g., [6,7,8,9,10,11]. From the view of effective dynamics [12], a minimal classical time crystal can be realized mathematically by including a potential with a welldefined regulator; it is characterized by a nonsmooth reversal of the velocity at the boundary. What is unusual is that the sudden velocity reversal is not caused by a “brick wall” potential; it is a consequence of spontaneous symmetry breaking analogous to the Higgs mechanism, but in the momentum space.
It is interesting and important to study this idea in the context of gravity. There are many important timedependent systems in general relativity (GR) such as the evolution of our universe, dynamical black holes, and gravitational waves. The timecrystal behavior of an oscillating scalar field in the expanding Friedmann–Lemaître–Robertson–Walker (FLRW) universe was constructed in [13,14,15]. The aim of this paper is to study the possibility of treating the universe itself as a time crystal. Such a cosmological model naturally depicts a cyclic universe, which would necessarily violate the nullenergy condition (NEC) in GR. Nevertheless, the idea of our universe being cyclic still attracts much attention, and the most famous model is the ekpyrotic universe, based on string theory [16]. The timecrystal mechanism provides an alternative realization.
Constructing cosmological time crystals was attempted in gravity on noncommutativity geometry [17]. In fact, the dynamics of classical time crystals [1, 12] can be easily realized in gravities, when we include higherorder curvature invariants. In this paper we consider Einstein gravity extended with appropriate Riemanncubic invariants, coupled to some homogeneous and isotropic perfect fluid. A new feature arising is that there is now a Hamiltonian constraint owing to the general diffeomorphism, whilst in a classical mechanical system the Hamiltonian yields an arbitrary conserved energy that should be bounded below.
In Sect. 2, we construct the cubic invariants that facilitate the timecrystal mechanism of [1, 12]. We study the properties of the resulting cosmological solutions. In Sect. 3, we present three explicit examples based on different types of the perfect fluid models. In the last example, we consider massive gravity instead of Einstein gravity. We discuss our results and conclude the paper in Sect. 4.
The theory and the cosmological model
A crucial ingredient in our construction involves the Riemanncubic invariants, which have in general eight terms:
In de Sitter (dS) or antide Sitter (AdS) spacetimes, these generate additional linear massive scalar and spin2 modes. Decoupling of the ghostlike spin2 mode requires (e.g. [18])
In this paper, we study cosmology in the FLRW metric
In the effective Lagrangian, the absence of both \({\ddot{a}}^2\) and \({\ddot{a}}^3\) terms requires
The first equation decouples the massive scalar mode. Intriguingly these are precisely the same conditions for the holographic atheorem [19]. For the linear \({\ddot{a}}\) term, we perform integration by parts so that the Lagrangian involves only the \({\dot{a}}\) term. With these conditions, we find that the effective Lagrangian is given by
The reason we end up with four parameters is that the cubic Euler density combination vanishes in four dimensions. We may restrict our consideration further to Ricci polynomials, namely \(7R^3 36 R\,R_{\mu \nu } R^{\mu \nu } + 36 R^{\mu }_{\nu } R^{\nu }_\rho R^{\rho }_\mu \) [20]. The analogous construction for quadratic invariants yields the Weylsquared term, which gives no contribution to the equations of motion for the FLRW metric.
We focus on Einstein gravity coupled to some perfect fluid with energymomentum tensor \(({{T^{a}}_b})_m = {\mathrm{diag}}(\rho _m,p_m,p_m,p_m)\), and the Riemanncubics. The Einstein equations yield
For simplicity, we consider an effective theory where the energy density is a function of the scale factor a. For example, the vacuum energy density is given by \(\rho _m=\Lambda _0\), the bare cosmological constant. In radiation or matter dominated universes, we have \(\rho _m\sim a^{4}\) and \(a^{3}\), respectively. For a free massless scalar \(\phi \), we have instead \(\rho _m\sim a^{6}\). If we express \(\rho _m=V(a)/(2a^3)\), the energymomentum conservation requires \(p_m= V'(a)/(6a^2)\), giving rise to the equation of state \(w={a V'}/(3V)\).
The equations of motion can now be derived from the effective Lagrangian
The negative kinetic energy \({\dot{y}}^2/2\) proposed in [1, 12] is hard to justify in classical mechanics, it arises naturally in gravity. This term should not be viewed as ghostlike owing to the general diffeomorphism, which imposes the Hamiltonian constraint
In fact this constraint is equivalent to the first equation in (2.6). By contrast, although H is conserved in classical mechanics, it does not necessarily vanish.
In this paper, we shall consider only \(\lambda >0\), such that the gravitational part of the Hamiltonian, \(H_0=6 a {\dot{a}}^2 + 2\lambda \, {{\dot{a}}^6}/{a^3}\), is bounded below. A key property is that, in terms of the canonical momentum
\(H_0(p,a)\) can be multivalued and the minimum of \(H_0\) occurs not when \(p=0\), but when
The \(p=0={\dot{a}}\) point is instead a local maximum. This is analogous to the Higgs mechanism, but in momentum space. \(H_0\) as a function of p is depicted in Fig. 1.
The evolution of a(t) near \(H_0^{\mathrm{min}}\) depends on the potential V(a). In order to study this behavior, we also examine \(H_0\) as a function of \({\dot{a}}\), depicted also in Fig. 1. The Hamiltonian constraint (2.8) at \(H_0^{\mathrm{min}}\) implies that
If it has no solution, (e.g. \(V=3a^3\), for a negative cosmological constant,) then \(H_0^{\mathrm{min}}\) can never be reached. If Eq. (2.11) has a solution at \(a=a_0\), the system may reach \(a_0\), but cannot stay there since \({\dot{a}}\ne 0\). Thus we must require that the cubic equation (2.8) for \(z={\dot{a}}^2\) has a positive root, and it does if and only if \({\widetilde{V}}(a)\le 0\) in the connected region. In fact, there are two positive roots, given by
The third root, corresponding to \(k=3\), is negative and should be ignored. In the vicinity of \(a=a_0\), we have
For matter satisfying the NEC (\(w\ge 1\)), we must have \(a\ge a_0\), thus a bounce occurs at \(a=a_0\). Furthermore, at \(a=a_0\), we have two vacua with \({\dot{a}}_0^\pm =\pm \lambda ^{1/4} a_0\), respectively, and they are energy degenerate, but they break the time reflection symmetry. The tunneling from one vacuum to the other keeps \(\dot{a}^2\) continuous, but causes \(\dot{a}\) to jump from the negative to the positive, or vice versa, analogous to the situation when a pingpong hits a brick wall.
It is instructive to determine whether there should be an external “brick wall” source, since \(\ddot{a}\), and hence the curvature, has a \(\delta \)function singularity of the comoving time. Assuming that the turning point \(a=a_0\) occurs at \(t=0\), we can solve the function a(t) at small t. The solution is smooth except at \(t=0\), and the extra source required for the bouncing behavior is formally given by
Since the energymomentum conservation does not involve a time derivative of p, we can effectively treat \(p_{\mathrm{ext}}=0\). It was demonstrated [12] that this matter source can be replaced by some welldefined regulator in classical time crystals, in which case it is of great interest to analyze the energy condition of the external source.
The physical picture is clear. Owing to the spontaneous symmetry breaking, the cosmology splits into two energydegenerated vacua, with \(\dot{a}_0^\pm \), respectively. As the universe shrinks to \(a_0\), it tunnels from the \(\dot{a}_0^\) vacuum to the \(\dot{a}_0^+\) one and starts to expand, creating a bounce at \(a=a_0\). It follows from (2.11) that this bounce mechanism is robust and will always occur for positive energy density.
Explicit examples

(I)
Bounce universes The simplest example is perhaps when w is a constant, for which \(V=2q^2 a^{3w}\) where q is a constant. For \(\lambda =0\), the universe is expanding with \(a=({\frac{3}{4}}(1+w)^2q^2 t^2)^{{1}/({3(1+w)})}\), with an initial spacetime singularity at \(t=0\). For nonvanishing \(\lambda \), a bounce must occur, at \(a_0^{3(1+w)}=\sqrt{\lambda } q^2/2\). The \(k=1\) solution is governed by
$$\begin{aligned} \dot{a}^2={\frac{2}{\sqrt{\lambda }}} a^2 \cos {\textstyle {\frac{\scriptstyle 1}{\scriptstyle 3} } }\left( \arccos \left[ \left( {\frac{a_0}{a}}\right) ^{3(1+w)} \right] +\pi \right) . \end{aligned}$$(3.1)Thus we see that, for the standard cosmology with positive energy density, the introduction the Riemanncubics generates a bounce universe. In Fig. 2, we plot the a(t) solutions for various w, taking \(a_0=1\). It is worth pointing out that our numerical analysis indicates that the solutions are stable against small perturbations of the initial condition \(a(0)=1\). Furthermore, the bounce scenario should be distinguished from those in the literature where \({\dot{a}}=0\) at the time of bounce. The consequence is that the NEC must be violated in the framework of Einstein gravity. In our case, \({\dot{a}}\) does not vanish at the time of bounce, but it tunnels from the negative value to the positive value in such a way that the matter system satisfies the NEC.

(II)
Cosmological time crystals If the potential V has a zero at finite \(a=A>a_0\); furthermore, \({\widetilde{V}}(a)<0\) for \(a\in (a_0, A)\), then a(t) shrinks smoothly from \(a=A\), until at \(a=a_0\) where it bounces, creating a cyclic universe. As a concrete example, we consider \(V=2\Lambda _0 a^3 + 6\alpha ^2 a\), where the second term can be generated by a sigma model [21]. For negative cosmological constant \(\Lambda _0=3/\ell ^2\), the potential vanishes at \(A=\alpha \ell \). In fact, for \(\lambda =0\), an exact solution can be found, namely \( a=A \sin (t/\ell )\). The solution appears to be cyclic, but the corresponding universe is not owing to the curvature singularity at \(a=0\). When \(\lambda \) is included, the \({\dot{a}}^6\) term has no effect at \(a=A\), where \(\dot{a}=0\). However, there is a turning point \(a_0\):
$$\begin{aligned} 0<a_0= {\frac{\sqrt{3}\,\alpha \ell \lambda ^{1/4}}{\sqrt{2\ell ^2 + 3 \sqrt{\lambda }}}}<A\,, \end{aligned}$$(3.2)where the universe bounces. The solution is depicted in Fig. 3. It should be pointed out that, for this particular model, the timecrystal mechanism is only possible for the negative cosmological constant, since \(\alpha ^2\) must be positive for the sigma model with the standard kinetic term [21]. Cosmological time crystals involving a positive cosmological constant will be presented next.

(III)
Cosmological time crystals from massive gravity In [12], time crystals typically have two jumping points. In our gravity model, for V(a) satisfying NEC, there can only be one, causing bounce of the universe. In order to reproduce the analogous behaviors of [12], we have to consider theory beyond Einstein. We find such a solution exists in dRGT (de Rham–Gabadadze–Tolley) massive gravity of [22] together with the Riemanncubics. The idea of describing the massive graviton was first proposed by Fierz–Pauli [23] in 1939. However, due to the Boulware–Deser (BD) ghost [24] of the interactions for massive spin2 fields in the Fierz–Pauli theory, few important developments had been made in the past decades, until the dRGT theory came along, which is a ghostfree realization of massive gravity and has attracted great attention in the general relativity community. We refer to, e.g., [25, 26] and the references therein for a comprehensive introduction of massive gravity. The Lagrangian of dRGT gravity together with the Riemanncubics is given by
$$\begin{aligned} \mathcal{L}= & {} \sqrt{g} \Big (R  2\Lambda _0 + m^2 ( \mathcal{U}_2 +c_3 \mathcal{U}_3 +c_4 \mathcal{U}_4 )\Big ) +\lambda \mathcal{L}_{{\scriptscriptstyle (3)}}\,,\nonumber \\ \mathcal{U}_2= & {} [\mathcal{K}]^2 [\mathcal{K}^2], \quad \mathcal{U}_3 = [\mathcal{K}]^3 3 [\mathcal{K}] [\mathcal{K}^2] +2 [\mathcal{K}^3] \,,\nonumber \\ \mathcal{U}_4= & {} [\mathcal{K}]^4 6 [\mathcal{K}^2] [\mathcal{K}]^2 + 8 [\mathcal{K}^3] [\mathcal{K}] +3[\mathcal{K}^2]^2 6[\mathcal{K}^4]\,,\nonumber \\ {\mathcal{K}^\mu }_\nu= & {} {\delta ^\mu }_\nu \sqrt{g^{\mu \lambda } \partial _\lambda \phi ^a \partial _\nu \phi ^b \eta _{a b}},\quad \eta _{a b} = \text {diag}(1,1,1,1)\,.\nonumber \\ \end{aligned}$$(3.3)\(\mathcal{U}_i\) are interaction potentials, and the rectangular brackets denote traces, e.g. \([\mathcal{K}] = \text {Tr}(\mathcal{K}) ={\mathcal{K}^\mu }_\mu \). For the corresponding Stückelberg fields \(\phi ^a = a_m (0, x_1, x_2, x_3)\), which was introduced to restore the diffeomorphism invariance [26, 27], the effective Lagrangian for the FLRW metric is given by (2.7) with
$$\begin{aligned} V= & {} 2 \Lambda _0 a^3 +6 m^2(a_m a )\nonumber \\&\times \Big ((4 c_3+4 c_4+2)a^2a_m (5 c_3 +8 c_4 +1) a \nonumber \\&+a_m^2 (c_3+4 c_4)\Big ), \end{aligned}$$(3.4)where \(a_m >0\). For a concrete demonstration, we choose
$$\begin{aligned} c_3={\textstyle {\frac{\scriptstyle 2}{\scriptstyle 87} } },\;\, c_4={\textstyle {\frac{\scriptstyle 7}{\scriptstyle 348} } },\;\, \Lambda _0={\textstyle {\frac{\scriptstyle 5989}{\scriptstyle 2900} } },\;\,\lambda =1,\;\, m={\textstyle {\frac{\scriptstyle 1}{\scriptstyle 10} } },\;\, a_m=10.\nonumber \\ \end{aligned}$$(3.5)
Note that, in this case, the cosmological constant \(\Lambda _0\) is positive here. The system has two turning points \((a_,a_+)=(1,3)\). The cosmology cycles between \(a_\pm \), as shown in the right plot of Fig. 2. Since we have \(a_\pm <a_m\) in this solution, there is no ghost excitation from the massive gravity sector [28].
Conclusions and discussions
Time crystals can arise when the timetranslational symmetry in the vacuum is spontaneously broken. A simple classical mathematical model [1, 12] involves a ghostlike kinetic \(\dot{y}^2/2\) augmented by the higherorder term \(\dot{y}^4/12\), such that the true vacuum is shifted down with nonvanishing velocity in a specific direction, hence breaking the time reflection symmetry. Such a system is hard to realize in classical mechanics, but it arises naturally in the effective Lagrangian in Einstein gravity on the FLRW metric, extended with appropriate higherorder curvature invariants.
We focused on a class of Riemanncubics and found that, for matter satisfying NEC, the timecrystal mechanism could generate bounces. The sudden change of the sign of \({\dot{a}}\) at the bounce is the effect of tunneling from one vacuum to the other while keeping \(\dot{a}^2\) continuous. Our analysis shows that this is a robust mechanism for bounce universes; it always occurs for positive energy density. Cyclic universes can also be constructed since shrinking a(t) from its maximum is consistent with NEC. We also considered massive gravity and constructed time crystals with two sudden reversing points.
Although we have restricted the Riemanncubics such that they do not generate linear ghosts in maximallysymmetric spacetimes, it is import to verify whether they may generate ghosts in our cosmological crystals. The general perturbation \(g_{\mu \nu }= {\bar{g}}_{\mu \nu } +h_{\mu \nu }\) for highorder gravities can be enormously complicated. We consider here the scalar perturbations \({h_\mu }^{\nu }\), which in Newtonian gauge is given by
The gravitational part of the linear equations of motion in the FLRW metric is given by
where \(\kappa \) is defined by \(\mathbf {\nabla }^2 \Psi = \kappa ^2 \Psi \) and \(\lambda \) is given by (2.5). We thus see that there are no higherorder time derivatives on \(\Psi \) in the equations of motion. This implies that the scalar perturbation of our cosmological time crystals can be ghostfree. In four dimensions, it is difficult to decouple the ghostfreedoms in higherorder gravities. The absence of the ghosts in the scalar perturbation in the FLRW model makes our cubic gravities interesting candidate for studying cosmology. There are, however, many questions that remain. For these higherderivative timecrystal models to be viable, it is necessary to investigate the unitary also for all the vector and tensor modes. Furthermore, it would be interesting to study whether there are ranges of model parameters for which the expanding phase occurs on the cosmological timescales of our universe. While the generalizations to higher dimensions and/or involving higherorder polynomial curvature invariants are straightforward, it is nevertheless of interest to work out the details, to see whether new phenomena could arise.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical study and has no experimental data associated to it.]
References
 1.
A. Shapere, F. Wilczek, Classical time crystals. Phys. Rev. Lett. 109, 160402 (2012). https://doi.org/10.1103/PhysRevLett.109.160402. arXiv:1202.2537 [condmat.other]
 2.
F. Wilczek, Quantum time crystals. Phys. Rev. Lett. 109, 160401 (2012). https://doi.org/10.1103/PhysRevLett.109.160401. arXiv:1202.2539 [quantph]
 3.
J. Zhang, P.W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I.D. Potirniche, A.C. Potter, A. Vishwanath, N.Y. Yao, C. Monroe, Observation of a discrete time crystal. Nature 543, 217 (2017). https://doi.org/10.1038/nature21413. arXiv:1609.08684 [quantph]
 4.
S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, V. Khemani, C. von Keyserlingk, N.Y. Yao, E. Demler, M.D. Lukin, Observation of discrete timecrystalline order in a disordered dipolar manybody system. Nature 543, 221 (2017). https://doi.org/10.1038/nature21426. arXiv:1610.08057 [quantph]
 5.
S. Autti, V.B. Eltsov, G.E. Volovik, Observation of time quasicrystal and its transition to superfluid time crystal. Phys. Rev. Lett. 120(21), 215301 (2018). https://doi.org/10.1103/PhysRevLett.120.215301. arXiv:1712.06877v5 [condmat.other]
 6.
P. Bruno, Comment on “Quantum Time Crystals”. Phys. Rev. Lett. 110(11), 118901 (2013). https://doi.org/10.1103/PhysRevLett.110.118901. arXiv:1210.4128 [quantph]
 7.
T. Li, Z.X. Gong, Z.Q. Yin, H.T. Quan, X. Yin, P. Zhang, L.M. Duan, X. Zhang, Spacetime crystals of trapped ions. Phys. Rev. Lett. 109, 163001 (2012). https://doi.org/10.1103/PhysRevLett.109.163001. arXiv:1206.4772 [quantph]
 8.
D.V. Else, B. Bauer, C. Nayak, Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016). https://doi.org/10.1103/PhysRevLett.117.090402. arXiv:1603.08001v4 [condmat. disnn]
 9.
N.Y. Yao, A.C. Potter, I.D. Potirniche, A. Vishwanath, Discrete time crystals: rigidity, criticality, and realizations. Phys. Rev. Lett. 118, 030401 (2017). https://doi.org/10.1103/PhysRevLett.118.030401. arXiv:1608.02589v3 [condmat.disnn]
 10.
K. Sacha, J. Zakrzewski, Time crystals: a review. Rept. Prog. Phys. 81(1), 016401 (2018). https://doi.org/10.1088/13616633/aa8b38. arXiv:1704.03735 [quantph]
 11.
N.Y. Yao, C. Nayak, L. Balents, M.P. Zaletel, Classical discrete time crystals (2018). arXiv:1801.02628 [hepth]
 12.
A. Shapere, F. Wilczek, Realization of “time crystal” Lagrangians and emergent sisyphus dynamics (2017). arXiv:1708.03348v1 [condmat.statmech]
 13.
J.S. Bains, M.P. Hertzberg, F. Wilczek, Oscillatory attractors: a new cosmological phase. JCAP 1705(05), 011 (2017). https://doi.org/10.1088/14757516/2017/05/011. arXiv:1512.02304 [hepth]
 14.
D.A. Easson, A. Vikman, The phantom of the new oscillatory cosmological phase (2016). arXiv:1607.00996 [grqc]
 15.
D.A. Easson, T. Manton, Stable cosmic time crystals (2018). arXiv:1802.03693 [hepth]
 16.
J. Khoury, B.A. Ovrut, P.J. Steinhardt, N. Turok, The ekpyrotic universe: colliding branes and the origin of the hot big bang. Phys. Rev. D 64, 123522 (2001). https://doi.org/10.1103/PhysRevD.64.123522. arXiv:hepth/0103239
 17.
P. Das, S. Pan, S. Ghosh, P. Pal, Cosmological time crystal: cyclic universe with a small \(\Lambda \) in a toy model approach (2018). arXiv:1801.07970 [hepth]
 18.
T.C. Sisman, I. Gullu, B. Tekin, All unitary cubic curvature gravities in \(D\) dimensions. Class. Quantum Gravity 28, 195004 (2011). https://doi.org/10.1088/02649381/28/19/195004. arXiv:1103.2307 [hepth]
 19.
Y.Z. Li, H. Lü, J.B. Wu, Causality and \(a\)theorem constraints on Ricci polynomial and Riemann cubic gravities. Phys. Rev. D 97(2), 024023 (2018). https://doi.org/10.1103/PhysRevD.97.024023. arXiv:1711.03650 [hepth]
 20.
Y.Z. Li, H.S. Liu, H. Lü, Quasitopological Ricci polynomial gravities. JHEP 1802, 166 (2018). https://doi.org/10.1007/JHEP02(2018)166. arXiv:1708.07198 [hepth]
 21.
W.J. Geng, H. Lü, Isotropic expansion of an inhomogeneous universe. Phys. Rev. D 90(8), 083511 (2014). https://doi.org/10.1103/PhysRevD.90.083511. arXiv:1407.0728 [hepth]
 22.
C. de Rham, G. Gabadadze, A.J. Tolley, Resummation of massive gravity. Phys. Rev. Lett. 106, 231101 (2011). https://doi.org/10.1103/PhysRevLett.106.231101. arXiv:1011.1232 [hepth]
 23.
M. Fierz, W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proc. R. Soc. Lond. A 173, 211 (1939). https://doi.org/10.1098/rspa.1939.0140
 24.
D.G. Boulware, S. Deser, Can gravitation have a finite range? Phys. Rev. D 6, 3368 (1972). https://doi.org/10.1103/PhysRevD.6.3368
 25.
K. Hinterbichler, Theoretical aspects of massive gravity. Rev. Mod. Phys. 84, 671 (2012). https://doi.org/10.1103/RevModPhys.84.671. arXiv:1105.3735 [hepth]
 26.
C. de Rham, Massive gravity. Living Rev. Relativ. 17, 7 (2014). https://doi.org/10.12942/lrr20147. arXiv:1401.4173 [hepth]
 27.
N. ArkaniHamed, H. Georgi, M.D. Schwartz, Effective field theory for massive gravitons and gravity in theory space. Ann. Phys. 305, 96 (2003). https://doi.org/10.1016/S00034916(03)00068X. arXiv:hepth/0210184
 28.
A. De Felice, A.E. Gumrukcuoglu, S. Mukohyama, Massive gravity: nonlinear instability of the homogeneous and isotropic universe. Phys. Rev. Lett. 109, 171101 (2012). https://doi.org/10.1103/PhysRevLett.109.171101. arXiv:1206.2080 [hepth]
Acknowledgements
We are grateful to YiFu Cai, YueZhou Li, Jiro Soda, ZhaoLong Wang and ZhangQi Yin for useful discussions. XHF, HH and HL are supported in part by NSFC grants nos. 11875200 and 11935009. SLL and HW are supported in part by NSFC grants nos. 11975046, 11575022, 11175016, 11947216, and China Postdoctoral Science Foundation 2019M662785.
Author information
Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP^{3}
About this article
Cite this article
Feng, XH., Huang, H., Li, SL. et al. Cosmological time crystals from Einsteincubic gravities. Eur. Phys. J. C 80, 1079 (2020). https://doi.org/10.1140/epjc/s10052020086223
Received:
Accepted:
Published: