Time crystals in primordial perturbations

Cosmological time crystal (TC) corresponds to a matter state where the periodic motion of field forms a limit cycle in its phase space. We explore what would happen if it existed in inflationary phase. It is found that the limit cycle responsible for TC will inevitably cause the periodic oscillation of the primordial perturbation spectrum. The oscillatory patterns of the spectrum depend on the TC parameters, and so encode the crystalline patterns of TC.

Introduction -Time crystals (TC) were put forward by Wilczek in 2012 [1] [2], which described the timeperiodic self-organized structures. In this new perspective, the spontaneous symmetry breaking occurs in the time-direction. Recently, the concept of TC has developed quickly in physical community, in particular condensed matter physics. The discrete TC has been discovered in the condensed matter experiments [3] [4].
In Ref. [5], based on the P (X, φ) theory, Bains, Hertzberg and Wilczek proposed the cosmological TC, where X = −(∇φ) 2 , which corresponds that a limit cycle should exist in φ,φ phase space, see FIG.1. However, such a cycle will inevitably cross the null energy condition (NEC) curve, which suggests the violation of NEC [5,6]. It is well-known that the violation of NEC will cause the instability of perturbation on short scales. Recently, this instability has be cured by applying the higher-order derivative operator of φ [7] (equivalently R (3) δg 00 , as in [8][9][10][11][12], where R (3) is the Ricci scalar on the 3-dimensional spacelike surface). Is such a stable cosmological TC observable? It is still an open question.
According to [5], the cosmological TC may appear in inflationary phase. If so, the time-periodic patterns of TC might be imprinted in the power spectrum of primordial perturbation. In our work, we will explore this possibility, and show that the limit cycle in φ,φ in phase space will result in the periodic oscillation of the primordial spectrum, which so encodes the crystalline patterns of TC, as in original concept [2]. Here, the TC model considered is a special subclass of G-inflation [13] [14].
A stable cosmological TC -In Ref. [5], the cosmological TC was built by applying such a Langrangian, with the coefficient c 0,1 = c 0,2 = 1, c 1,1 = β > 0, and the Higgs-like potential where β > 0 and α > 0 are the parameters, both impact the period of cosmological TC [5,6]. Here, although the Lagrangian (1) is that of k-inflation [15,16], the inflation is actually potential-driven, and Λ = const. is responsible for the inflation. We have We solve the motive equation of φ numerically and plot it in FIG.1. We see that φ oscillate around the maximum point of its potential. As showed in Ref. [5], such solutions are the attractors, which form the limit cycles in φ,φ phase space. The amplitude and period of oscillations keep constant, such a oscillatory matter state is called the cosmological TC.   In the phase diagram, the yellow one is the limit cycle. Inside the green curve is the area of violating NEC. Inside the red ellipse, the background field are ghost. The gray and brown curves are trajectories of background fields in phase space. At the beginning, the gray one is inside the limit cycle and the brown one is outside the limit cycle.
where Tµν is the energy-momentum tensor. The boundary of violating the NEC is another elliptic curve PX = 0 , that isφ we use a green line to show this curve in the phase space. In some discrete pionts ϕ = 0, conservation of energy giveφ in terms of the piontφ * ϕ = ± 2 3 ( 1 + The evolution of background field φ, its Higgs-like potential and the limit cycle. Here, the Planck mass MP ∼ 10 3 . In φ,φ phase space, the yellow one is the limit cycle, which is the attractor [5]. See the gray and brown curves for the trajectories of φ with different initial φ0,φ0 . Inside the green curve, the NEC is violated. Inside the red ellipse, the field is ghost-like. The existence of limit cycles requires that the NEC must be violated, because the sign ofḢ will change periodically. The NEC condition for any null vextors k µ is arXiv:1907.09148v2 [gr-qc] 27 Dec 2019 FIG.1(c), the boundary of NEC violation is an elliptic curve P X = 0, which isφ 2 + βφ 2 = 1, where P = n,m =0 c n,m φ 2n (X/2) m , see the Lagrangian (1). Another elliptic curve is ρ X = 2XP XX + P X = 0, which is 3φ 2 + βφ 2 = 1. Inside it, ρ X < 0. Because the limit cycle will not cross the ellipse with ρ X = 0, but must cross the boundary of NEC violation, the bound on the oscillating amplitude of the field is [ which shows how the width of the limit cycle depends on the parameters α and β.
It is well-known that if the NEC is violated, the Langangian (1) will be instable in perturbative level, since c 2 s < 0 is inevitable, c s is the sound speed of perturbation. Here, we will cure it by considering Galileon field operator ∼ φ.
It is simple to work in unitary gauge φ = φ (t). The EFT Lagrangian of cosmological perturbation corresponding to the Lagrangian (1) with the Galileon operator ∼ φ is where and the coefficients is free, K is the extrinsic curvature on the 3-dimensional spacelike surface, φ is related to δK and φ σ = ∇ σ φ, see, e.g.Ref. [17].
The action of scalar perturbation ζ for (5) is where Here, the stabilities of perturbations require both Q s > 0 and c 2 s > 0.
According to (10), without the operator δKδg 00 , we have When the NEC is violated,Ḣ > 0. Thus we have else c 2 s < 0, or Q s < 0, the corresponding evolution is instable. Considering the operator δKδg 00 , we have Now, if one properly adjusts the coefficient m 3 3 of δKδg 00 , it is possible that both Q s > 0 and c 2 s > 0 are satisfied simultaneously. Thus the operator δKδg 00 may be applied to cure the instabilities of scalar perturbations in cosmological TC.
Here, it is convenient to assume that m 3 3 is proportional to H, m 3 3 = xM 2 P H (x = 0). According to (9) and (10), we have noting thatφ 4 P XX is far smaller than H 2 during inflation. We keep c 2 s ≈ 1, so get x = 1/2 (which brings Q s ≈ M 2 P /3 > 0). We plot c 2 s in FIG.2. The blue line is c 2 s without the operator δKδg 00 . In contrast, the magenta line is that with δKδg 00 . Due to the existence of limit cycle, both H andḢ are oscillating. Thus although c 2 s ≈ 1, it is affected by ,φ 4 P XX /H 2 and is also oscillating. In certain sense, the oscillations of both H and c 2 s actually encode the crystal shape of cosmological TC.
Oscillatory patterns of power spectrum -The power spectrum of primordial perturbation is given by In the momentum space, the motive equation of ζ k is During the inflation, | | = |Ḣ/H 2 | 1, but |Q s QsH | 1 is not always valid. Thus we will solve Eq.(15) numerically.
This suggests that the initial state for u k may be the Bunch-Davies state u k = e −ikη √ 2k . In physical time, it equals to We plot the perturbation spectrum P ζ (k) in FIG.3 for different limit cycles. As expected, P ζ (k) is oscillating periodically, which is actually an inevitable result of the limit cycle existing in φ,φ phase space. It is interesting to search for the possible oscillating patterns of power spectrum P ζ (k) for different parameters α and β, which control the shapes of limit cycles. As found in Ref. [5], the larger β is, the larger allowed range of α is. But a bigger β will bring a shorter minor When α increases, the "ripple" in the bottom of spectrum will move up, and after crossing the peak, it will descend. When α is fixed and β is changed, the case is reverse. Thus the shape of limit cycle (equivalently the crystalline patterns of cosmological TC) will be encoded in the different oscillating patterns of power spectrum. Discussion -Recently, as a special matter state, the cosmological TC has been implemented stably. How to look for such a matter state in cosmological observations is a significant issue.
In our work, we explored what would happen if the cosmological TC existed in the inflationary phase. We apply the operator m 3 3 (t)δKδg 00 (∼ φ) to remove the instability relevant with c 2 s < 0, and calculated the spectrum of corresponding primordial perturbation. It is found that the limit cycle responsible for TC in φ,φ phase space will inevitably cause the oscillation of spectrum. We point out that the oscillatory patterns of the primordial spectrum depends on the TC parameters, and so encode the crystalline patterns of TC. Thus if the cosmological TC existed in inflationary phase, the imprint of TC might be preserved in the CMB. It will be interesting to search for the corresponding signals in Planck data. There would not be an inflationary stage forever. TC will dissolve due to interactions with a 'waterfall' field σ [25]. During inflation phase, field σ is stabilized at its only minimum where σ = 0. When φ is smaller than the effective mass, the 'waterfall' field σ becomes tachyonic and ends inflation. Another field coupled with φ may offer a large no-gaussianity which can be observable in the feature experiment. It is a new view to observe cosmological TC. In addition, adding external matter can have very strong implications for stability of models with nonstandard kinetic terms where the perturbations of φ near the ghost region c 2 s < 0 [26]. However, in our model, the m 3 3 (t)δKδg 00 term will stay the perturbations far away from the ghost region, so it is still a stable model.
It is well-known that the inflation scenario itself can not avoid the initial singularity. The cosmological TC has been also investigated in certain theories of modified gravity [27][28][29], in particular Ref. [7] with the operator R (3) δg 00 . This might provide a link of the cosmological TC to the nonsingular cosmologies, which is worthy of deeply exploring.

ACKNOWLEDGMENTS
We thank Gen Ye and Yong Cai for helpful discussions. This work is supported by NSFC, Nos.11575188, 11690021.

APPENDIX
In this paper, we set units c = = 1 and the sign (− + ++).