Introduction

Dark energy is unquestionably the most mystifying energy ingredient in the universe. While many experiments are designed to measure its time evolution, and many theoretical models have been devoted to understanding dark energy, its nature at the fundamental level, remains obscure.

Previously, in1 we investigated a particular model2 of time crystals from a cosmological point of view and showed that it produced phantom dark energy in the universe. Time crystals arise when a system described by a time independent Hamiltonian has stable bound state solutions that spontaneously break the time symmetry of its Hamiltonian and reduce its time translation symmetry to a discreet subgroup that is periodic in time. The crystalline type structures formed in this manner, coined time crystals, oscillate in time rather than space.

In this letter, we investigate the general case, namely whether scalar field models of phantom dark-energy are made up of time crystals and, in turn, if a fluid of time crystals always behaves as phantom dark-energy. We claim that this relation between phantom dark energy and time crystals holds in general.

If this is indeed the case, it would shed light on the nature of phantom dark energy and imply that fundamentally it is an ideal fluid of time crystals interwoven in the fabric of spacetime.

Phantom dark energy

To prove this claim, we first investigate the conditions on a scalar field required by phantom dark energy.

Let us begin with a generalized Lagrangian on an Friedman Robertson Walker (FRW) background with line element \(ds^2 = - dt^2 + a(t)^2 dx_{(3)}^2\),

$$\begin{aligned} {\mathcal {L}} = f(\phi ) g(X) - V(\phi ), \end{aligned}$$
(1)

where \(f(\phi )\), \(V(\phi )\) are positive semi-definite functions of the scalar field \(\phi\); \(X = g_{\mu \nu } \frac{\nabla ^{\mu }\phi \nabla ^{\nu }\phi }{2}\). In what follows, we consider the homogeneous mode \(\phi (t)\) which, as is typical of inflationary models, is a function of time. At the end, we briefly turn our attention to the role of perturbations \(\delta \phi (t,x)\) on the stability of these models. For \(\phi (t)\), \(X\) reduces to \(X= \frac{\dot{\phi }^2}{2}\) and \(g(X)\) is a function of X.

We can recover the canonical case of an ideal fluid by taking \(f(\phi )=1, g(X) =X\). However, we show next that canonical Lagrangians cannot produce phantom dark energy.

For the ideal fluid given by Eq. (1), the pressure of the fluid \(p\) is given by the Lagrangian \({\mathcal {L}} = p\) of the scalar field, and its energy density \(\rho\) by the respective Hamiltonian of the field, \({\mathcal {H}} = \rho\). The transformation between the Lagrangian and the Hamiltonian is obtained by:

$$\begin{aligned} {\mathcal {H}} + {\mathcal {L}} = \Pi \dot{\phi } \end{aligned}$$
(2)

where the conjugate momentum is

$$\begin{aligned} \Pi = \frac{\partial {\mathcal {L}}}{\partial \dot{\phi }} = f(\phi ) g'(X) \dot{\phi }\, . \end{aligned}$$
(3)

Therefore,

$$\begin{aligned} {\mathcal {H}} = f(\phi ) ( 2 X g'(X) - g(X) ) + V(\phi ) \end{aligned}$$
(4)

with the Lagrangian of the scalar field \({\mathcal {L}}\) given by Eq. (1), where prime denotes \(\frac{\partial }{\partial X}\). We require the energy of the scalar field given by the Hamiltonian of Eq. (4) to be positive, \(\rho>0\).

Phantom behavior demands the equation of state of the ideal fluid to be: \(w_{\phi } \le - 1\), where

$$\begin{aligned} w_{\phi } =\frac{p}{\rho } = \frac{f(\phi ) g(X) -V(\phi )}{f(\phi ) (2X g'(X) - g(X) ) +V(\phi )} \end{aligned}$$
(5)

which is equivalent to demanding \(\rho + p \le 0\).

Since \(\rho + p= f(\phi ) 2 X g'(X)\), a phantom dark energy fluid that satisfies \(w_{\phi } \le -1\) leads to condition \(g'(X) < 0\) given that \(X \ge 0\).

The latter implies that for a phantom dark energy fluid, the function \(g(X)\) must be a decreasing function of \(X\), which proves that \(g(X)\) cannot belong to the class of canonical Lagrangians that are linear in \(X\), i.e., \(g(X) \ne X\).

Furthermore, the special case of \(g(X)\) which is linear in \(X\) but with the ’wrong’ negative sign, \(g(X)= - \kappa X\), where \(\kappa>0\) is a parameter, is also excluded because it is unbounded from below leading to unbounded negative energies2,3,4, which cannot give rise to a Friedmann-Roberson-Walker (FRW) universe. In the latter case, there are some counter examples of modified or scalar tensor gravity inspired models, with negative kinetic energies that are stable self accelerating cosmologies, e.g5. (It is straightforward to see that a rotation of the field \(\phi -> i\psi\) would recover the canonical case \(g(X) = \kappa \dot{\psi }^{2}/2\), that we ruled out as a candidate for phantom dark-energy models).

Einstein field equations relate the expansion of the universe to the dark energy fluid pressure \(p\) and energy density \(\rho\) such that, modulo prefactors of \(8\pi G/3\), the Hubble parameter is \(H^{2} =(\frac{\dot{a}}{a})^2 \simeq \rho\) and \(\dot{H} \simeq - (\rho +p)\), relations that rule out scalar field models with unbounded negative energies of the field discussed above.

The Bianchi identity \(\dot{\rho } + 3H (\rho +p)=0\) is a combination of the above two Einstein’s equations. It is equivalent to the field’s equation of motion, namely:

$$\begin{aligned} \dot{\phi } \left[ \dot{\Pi } + 3 H \Pi + \frac{\partial V(\phi )}{\partial \phi } - g(X) \frac{\partial f(\phi )}{\partial \phi }\right] = 0 \end{aligned}$$
(6)

with \(\Pi\) the conjugate momentum of the field in Eq. (3). This equation can be satisfied by the trivial solution \(\dot{\phi }=0\), or when the squared bracket equals zero, that is, by solving the equation for conservation of momentum, the function \(\Pi (X)\). Given a knowledge of \(V(\phi )\) and \(f(\phi )\), Eq. (6) can be solved numerically. However, it should be noted that for phantom dark energy the Hubble parameter grows rapidly with time and soon overwhelms the last two unspecified potential energy terms, a point that will be important to discussing the role of potential terms below.

This set of equations, along with the requirement that its energy density be positive, place constraints on the phantom dark energy stress energy tensor. (In principle, we could include other matter and radiation contributions to the above stress energy components that enter Einstein’s equations, however as is well known6,7 those components dilute away fast and phantom dark energy dominates the expansion).

For these reasons, well-behaved stable phantom dark energy models belong to the class of non-canonical Lagrangians which, in addition to the linear term, contain higher-order terms in powers of \(X\), meaning \(g''(X) \ne 0\). In fact, since the requirement for phantom behavior \(w_{\phi } \le -1\) leads to \(g'(X) \le 0\) that condition places a constraint on the second derivative \(g''(X)\ge 0\) for the energy to be bound from below and for the theory to be stable.

Time crystals

As shown above, the class of phantom dark energy models with the non-canonical kinetic term \(g(X)\) is such that it generically contains higher powers of \(X\) besides the linear term in \(X\), that is \(g''(X)\ne 0\). In fact, we further know that since \(g'(X) < 0\) for \(w_{\phi } < -1\) then \(g''(X)>0\). Hence, the Hamiltonian \({\mathcal {H}}\) of Eq. (4) which provides the energy density of the non-canonical fluid, obtained from the Lagrangian \({\mathcal {L}}\) will be related to the Lagrangian in a non-smooth way, given the above conditions on the function \(g(X)\), and the expression of conjugate momentum, Eq. (3), \(\Pi = \dot{\phi } g'(X) f(\phi )\).

Since \(\Pi\) is a nonlinear function of \(\dot{\phi }\) and of \(g(X)\) which has linear and quadratic terms of opposite signs, (and possibly higher order powers of \(X\)), then it can be seen from the expression of momentum Eq. (3) and of the Hamiltonian Eq. (4), that \({\mathcal {H}}\) will be a multi-valued function of (\(\dot{\phi }\), \(\phi\)) indicating the existence of orbits, with solutions \(\dot{\phi }\ne 0\), along which the kinetic energy is minimized.

Let us show explicitly the existence of orbits and highlight the fact that the non-smooth relation between the Hamiltonian and the Lagrangian which is a consequence of the constrained non-canonical kinetic function \(g(X)\), breaks the time translation symmetry, i.e. these turning points bounding the orbit of the crystal are found at \(\dot{\phi }_{t} \ne 0\).

Kinetic energy is minimized at

$$\begin{aligned} \frac{\partial {\mathcal {H}}}{\partial \dot{\phi }} = \frac{\partial {\mathcal {H}}}{\partial X} = 0\,. \end{aligned}$$
(7)

This equation is equivalent to finding cusps of conjugate momentum\(\Pi\)

$$\begin{aligned} \frac{\partial \Pi }{\partial \dot{\phi }} =0 \, \end{aligned}$$
(8)

at \(\dot{\phi } = \dot{\phi }_{t} = \pm \sqrt{X_{t}}\). They result in the following constraint equation at \(\dot{\phi } =\dot{\phi }_t\):

$$\begin{aligned} g'(X) + 2 X g''(X) = 0\, \end{aligned}$$
(9)

The turning points \(\phi _{t}, X_t\) are found by solving Eq. (9). Functions \(g(X)\) and \(\Pi\) are non linear functions of \(\dot{\phi }\) with the first term \(g' <0\) and the second term \(g''>0\) in Eq. (9). Hence, due to the constraints on the nonlinear structure of the function \(g(X)\), solutions given by the cusps of the momentum function which minimize energy will be found at nontrivial points \(\dot{\phi } \ne 0\) thereby indicating motion of the bound state that breaks time translation symmetry, yielding a time crystal solution.

We came across a specific example of time crystals producing phantom dark energy in1. Another illustration that satisfies the conditions on \(g'(X), g''(X)\) would be a kinetic term of the form \(g(X)= e^{- \alpha X}\) which, according to Eq. (9) gives the turning points at \(\dot{\phi }_{t} = \pm \sqrt{X_{t}}= \pm \sqrt{1/2\alpha }\). In general we can expand any nonlinear function as a series expansion \(g(X)= \Sigma \alpha _{n} X^n\) for \(n =1,2,3,4...\) with coefficients \(\alpha _n\) constrained by the requirements \(g'(X)<0 , g''(X)>0\) and obtain the turning points from the cusps of the momentum. In this case, the solutions to Eq. (9) are \(X_{t} = |\alpha _{n}|/2(n+1)|\alpha _{n+1}| \ne 0\).

In the general case, independently of the potential \(V(\phi )\), solutions of Eq. (9) at the cusps of the momentum with \(\dot{\phi }\ne 0\) are bound states that minimize kinetic energy bound by the turning points at \(\dot{\phi } = \pm \dot{\phi _t} = \pm \sqrt{2X_t} \ne 0\) of the time crystal. In sum, the same conditions placed on phantom dark energy require a non-canonical multi-valued Hamiltonian \({\mathcal {H}}\) and nonlinear expression of the momentum \(\Pi \simeq \dot{\phi }g'(X)\) also imply that the minima of these expressions will also occur at nontrivial points \(\dot{\phi _t} \ne 0\). The fact that the cusps of the conjugate momentum \(\Pi\) for this multi-valued energy function \({\mathcal {H}}\) of \(\dot{\phi }\) correspond to the boundary of the time crystal structure in field space becomes apparent from noticing that there is motion \(\dot{\phi } \ne 0\) of the crystal as a whole that breaks the time translation symmetry.

We have seen until now that the potential energy \(V(\phi )\) has not played an important role in our discussion of phantom dark energy or time crystals since orbits of the time crystal are found by minimizing kinetic terms. Yet, the Bianchi identity which give momentum conservation Eq. (6) depends on the potential energy term. Here we would like to understand whether our results are completely general for any choice of potential energy or whether the potential needs to be fine tuned to specific cases. (For the sake of simplicity, we explore this question by setting \(f(\phi )=1\) without losing generality, it can always be factored out by redefining the potential \(\tilde{V}(\phi ) = V(\phi )/f(\phi )\)).

From the Lagrangian equation of motion or equivalently the Bianchi identity \(\dot{\Pi }= \frac{\partial {\mathcal {L}}}{\partial \phi } - 3 H \Pi\) which simplifies to \(\dot{\Pi } + \frac{\partial V(\phi )}{\partial \phi } + 3 H \Pi =0\), where \(H\) is the Hubble parameter. It should be noted that in presence of the coupling of the scalar field to gravity and an expanding universe, the field equation of motions and perturbations(see6,8,9) are vastly different from those in2,3. In a phantom dark energy universe, the Hubble parameter grows with time as \(H\simeq a(t)^{-\frac{3}{2} (1+w_{\phi })}\), where \((1+w_{\phi })\le 0\) and \(a(t) = a(t_0) [ - w_{\phi } + (1+w_{\phi }) t/t_0 ]^{\frac{2}{3(1+w_{\phi })}}\) is the scale factor, with \(t_0)\) the present time. Clearly, the scale factor diverges within a finite time \(t \simeq t_{0} w_{\phi }/(1+w_{\phi })\) from present day \(t_0\).The Hubble friction term quickly dominates over the potential energy term \(\frac{\frac{\partial V(\phi )}{\partial \phi }}{3H} \simeq \frac{1/a(t)^2}{3 a(t)^{-3(1+w)/2}} \rightarrow 0\) for any reasonable potential rendering the potential term insignificant, effectively driving it to a minimum, in the equation of motion. In this sense, the scalar field enters an attractor behavior due to the Hubble drag term, (see9 for further discussion).

The term ’reasonable potentials’ is used here to denote potentials\(V(\phi ), f(\phi )\) that satisfy the positive energy condition \(\rho \ge 0)\), which is achieved fo r \(g(X_{t}) \le \frac{V(\phi )}{f(\phi )}\). The latter restriction on the class of potentials is very weak since the kinetic function \(g(X)\) is bound to take values within the time crystal region at \(X\le X_t\) and one can always rescale the potential by an overall constant factor to satisfy the inequality.

Therefore, independently of the chosen functional form of the potential, and owing to a fast growing Hubble drag term which overwhelms the potential term, in general, non canonical time crystal models, with orbits that minimize kinetic energy and have motion \(\dot{\phi } \ne 0\) which breaks the underlying symmetry, will produce phantom dark energy.

Stability of time crystals

We need to show that the region of time crystals bounded by the turning points, given by the cusps of the conjugate momentum, corresponds to stable bound states.

Before time crystals were discovered, authors of4 who studied this class of Lagrangians in the context of k-essence naturally interpreted the boundary of the time crystal with a terminating singularity since, as we will see next, the speed of sound diverges there. They relied on Einstein equations, to conclude from Friedman’s equation and the diverging speed of sound at the boundary, that the expansion of the universe terminates abruptly when the field reaches its boundary value \(\phi _t\).

In fact, as shown a few years later in2,3,10 when time crystal solutions were discovered, time crystals are stable bound state solutions and their boundary at \(\phi =\phi _t\) behaves like a brick wall potential for the field, namely as the field approaches its boundary, the speed of sound squared diverges, implying that the field simply reverses its momentum there while all the time conserving the energy thus preserving the bound state. In this work, the phantom dark energy fluid is collections of these bound states, of the time crystals, and we show next that the expansion of the universe does not suffer a terminating singularity. Rather, it continues into an accelerated expansion until doomsday: the Big Rip.

Let’s estimate the speed of sound squared and show that \(C_s^{2}>0\) everywhere in the time crystal field region with \(\phi \le \phi _t\) and it diverges at the boundary \(\phi _t\). In other words, the bound state structure is stable against small perturbations and the field cannot break the crystal structure to jump over values higher than its orbit \(\dot{\phi _t}\) without investing a large amount of work and energy to break the crystal-bound state, because at the boundary the speed of sound squared \(C_s^2 -> \infty\) at \(\dot{\phi } -> \dot{\phi _t}\). At the boundary, the field bounces back by reversing its momentum and remains confined within the well-behaved bound state solutions. The diverging speed of sound at the boundary simply serves to reinforce the stability of the crystal by keeping the field bound within the crystal regime despite the accelerated expansion of the universe, discussed in the context of the field’s equation of motion above.

Actually, since the fluid is a collection of bound state crystals, we expect to find the speed of sound to be not only positive but to be \(C_{s}^{2} \ge 1\) characteristic of the crystalline structures. If such a speed of sound is found observationally, it would be an indicator of a crystalline type of dark energy. The relation of the speed of sound to background perturbations, implies that gravitational collapse of background perturbations into dark matter will be highly dampened, and none of the symmetries, such as Lorentz invariance, are violated, as explained in detail in6,9.

The sound speed squared \(C_s^2\) for an ideal fluid with a stress energy tensor

$$\begin{aligned} T_{ab} = \rho u_{a} u_{b} + g_{ab}(\rho + p) \end{aligned}$$
(10)

with \(u_a\) the four vector normalized to unity and \(g_{ab}\) the metric of the FRW universe, is defined as:

$$\begin{aligned} C_s^2 = \frac{p'_X}{\rho '_X} = \frac{\mathcal {L'}_X}{\mathcal {H'}_X}. \end{aligned}$$
(11)

where the subscript \(X\) indicates derivatives with respect to that variable. Given the Lagrangian and Hamiltonian expressions in Eq. (1) and Eq. (4), in the general non-canonical case the speed of sound is:

$$\begin{aligned} C_s^{2} = \frac{g'(X)}{ ( g'(X) + 2 X g'' (X) )} \end{aligned}$$
(12)

The stability condition requires that the speed of sound is positive, \(C_{s}^{2}>0\). The latter, in combination with the condition from the phantom equation of state \(g'(X)\le 0\), and from Eq. (12) implies that \(g'(X) + 2X g''(X) \le 0\). Given the stability condition of the fluid, \(g''(X)>0\), and phantom behavior \(g'(X) <0\) naturally results not only in the speed of sounds being positive but, furthermore, in \(C_{s}^{2} \ge 1\), as can be seen from Eq. (12).

Note from the Hamilton equations, Eq. (6), that the exact same condition that minimizes kinetic energy \(\mathcal {H'}_X =0\) at \(\dot{\phi } = \pm \dot{\phi }_t\) and gives the turning points of the time crystals, also enters in the denominator of the expression for the speed of sound squared, Eq. (11). Therefore, given that the nominator \(\mathcal {L'}_{X} < 0\) for the phantom dark energy fluid (\(g'(X) < 0\) in Eq. (12)), it can be clearly seen that the speed of sound squared will diverge at the boundary as expected

$$\begin{aligned} C_s^2 =\frac{\mathcal {L'}_X}{\mathcal {H'}_X} = \infty \quad \textit{at}\quad \dot{\phi } =\pm \dot{\phi }_t \end{aligned}$$
(13)

Since\(\mathcal {H'}_X =0\) is the requirement for minimizing the energy along the orbit and it defines the time crystal, with turning points at \(\dot{\phi }_t \ne 0\) which break the time translation symmetry, a consequence of the fact that the crystal has motion as a whole. The turning points \(\phi _t\) with the diverging speed of sound squared define a brick wall in field space, an upper bound for the dark energy fluid defined by the non-canonical field \(\phi\) which is confined to take values within its bound state, the time crystal. Energy is conserved at the turning points as the field reverses momentum, therefore the expansion of the universe does not terminate abruptly. Instead the field is forced to oscillate within the time crystal region.

A numerical study of potential observational signatures, including metric perturbations, requires selecting a class of models and it will be presented in a forthcoming paper. Hence, we can draw some general remarks regarding perturbation around time crystals. For simplicity, we will keep the metric perturbations frozen and set \(f(\phi ) =1\). Then, an approximate analysis of the scalar field perturbations \(\delta \phi (t,x)\) coupled to gravity on an accelerated expanding FRW background, with Hubble parameter \(H\), reveals stability, in part due to the periodic nature of the crystals as we show below, as well as the positive speed of sound squared and the dominant role of the Hubble drag term. For the cosmic fluid made of time crystals, where the Hamiltonian is identified with the energy density, \(\rho = {\mathcal {H}}\) and pressure with its Lagrangian \(p= {\mathcal {L}}\), the perturbation equation to first order is

$$\begin{aligned}&\delta \ddot{\phi } - \frac{C_{s}^{2}}{a^2} \delta \phi _{,ii} + 3H \delta \dot{\phi }\nonumber \\&\quad +\left( 3 \frac{g''}{g' + 2 X g''} \ddot{\phi } \dot{\phi }+ \frac{g'''}{g' + 2 X g''}\ddot{\phi }\dot{\phi }^3 \right) \delta \dot{\phi }\nonumber \\&\quad + \frac{V_{\phi \phi }}{g' + 2X g''} \delta \phi =0 \end{aligned}$$
(14)

where the index \(\prime\) mean derivative with respect to \(X\) and\(i\) derivative with respect to the spatial variable \(x,y,z\). (Notice that the denominator in the last three terms is simply \(\rho '_{X} = g' + 2Xg''.\)) Rewriting it in term of Fourier modes, the second term in Eq.(14) becomes \(-\frac{C_{s}^{2} k^{2}}{a^2} \delta \phi\), where \(C_{s}^2\) is the speed of sound squared which is positive. Due to the periodic nature of the crystal terms, those terms containing odd power of \(\dot{\phi }\) average to zero, that is \(<\dot{\phi }> \simeq 0 , <\dot{\phi }^3> \simeq 0\), therefore the terms in the round brackets in Eq.(14) average to zero. Therefore Eq. (14) becomes

$$\begin{aligned} \delta \ddot{\phi } + 3H \delta \dot{\phi } + \frac{C_{s}^{2} k2}{a^2} \delta \phi + \frac{V_{\phi \phi }}{g' + 2 Xg''} \delta \phi \simeq 0\end{aligned}$$
(15)

Specific conclusions require a specific model. However it is possible to draw some general conclusions. For example, the Hubble drag term dominates over the potential term for any reasonable potential, as shown in the previous section counting powers of \(\frac{1}{a(t)}\) in their ratio; the odd powers of \(\dot{\phi }\) average to zero due to the oscillating nature of the crystal, simplifying Eq.(14); while \(\rho _X\) will average to some finite number given by the parameters of the specific crystal model. Besides, the crucial factor for stability of any model is the speed of sound squared being positive. The solution to the above simplified equation Eq. (15) is the familiar under-damped decaying pendulum solution.

To remove any concern, about the longest super-horizon wavelengths, where the small \(V_{\phi \phi }\) term may play a negative role on stability, let us now look at the perturbation energy to second order,

$$\begin{aligned} \delta ^{2}E = \frac{\rho _X}{2}\left[ \delta \dot{\phi }^2 + \frac{C_{s}^{2} k^2 }{a^2} \delta \phi ^2 + \frac{V_{\phi \phi }}{\rho _X} \delta \phi ^2 \right] \end{aligned}$$
(16)

With all the caveats of the Hubble drag term dominating, replacing an approximate decaying solution from the perturbations equation (\(\delta \phi (t,x) \simeq e^{-\gamma t -i k x} cos(\Gamma t)\) where \(\gamma , \Gamma\) are given by a combination of time dependent \(H, C_{s}^2,\) and \(m^2 \simeq V_{\phi \phi }\), when solving Eq. (15)), the rate of change of perturbations energy of (16) over time is roughly \(d (\delta ^{2}E)/dt \simeq - 6 H \delta \dot{\phi }^2 <0\) which is negative showing that no energy is taken from the homogeneous mode energy, \(\rho\).

Conclusions

In this letter we demonstrated that the conditions placed on a scalar field with a time independent Hamiltonian corresponding to the phantom dark energy fluid are equivalent to those that produce time crystals - stable bound states with motion in time \(\dot{\phi }\ne 0\) which break the underlying time symmetry of the Hamiltonian. We argued on general grounds that the potential \(V(\phi )\) is overpowered by the Hubble drag term in the field’s equation of motion and does not play an important role in the conditions of a phantom dark energy equation of state \(w_{\phi } \le -1\) or in its speed of sound squared. This means that as long as the conditions for a phantom fluid are satisfied, time crystals that make up the phantom dark energy fluid will always exist, independently of the particular scalar field model.

The Hamiltonian equation for minimizing energy defined the time crystal orbit. These energy conditions show that the same structures which are stable along the orbit of the crystal, break time symmetry and minimize energy, with a spike \(C_s^{2} -> \infty\) at the boundary, defining a brick wall in field space where the field reverses its momentum and is contained within the stable bound state regime, but a finite \(C_s^2 \ge 0\) everywhere else within the time crystal region. Simultaneously, they give rise to the phantom dark energy fluid equation of state on the background spacetime, satisfying the conditions \(w_{\phi }<-1\) and the need for a non canonical Lagrangian for the phantom fluid.

Given the nature of phantom dark energy discussed in this letter, we expect the propagation of perturbations to have \(C_s^2 \ge 1\) for the phantom dark energy made up of time crystals, typical of crystalline structures11. The latter contributes a term in the perturbation equation11,12,13 which leads to the suppression of structures and dampening of the gravitational instabilities that would have produced cold dark matter, see6,14.

In conclusion: we have shown that phantom dark energy fluid produces time crystals and conversely time crystals give rise to phantom ideal fluids. The two are equivalent for generic case non-canonical models.

If dark energy in our universe turns out to be of a phantom type, then whenever we look at our sky we glimpse at a spacetime threaded with times crystals scintillating simultaneously throughout the universe and beaming with phantom dark energy. As the volume of the universe expands and races rapidly towards the Big Rip, every instant the fabric of our spacetime is being suffused with ever more time crystals.

Furthermore, the unique signature on the propagation of perturbations through the speed of sound squared, \(C_{s}^2>1\) which are expected from crystalline structure, although it doesn’t violate Lorentz invariance and causality as discussed in6,14, it can still give rise to testable observations in the CMB sky such as: shifting of the baryonic acoustic peaks; damped perturbations that contribute to cold dark matter sector; signatures on the ISW region of CMB’s low multipoles, and possibly on \(\sigma _8\). We explore these possibilities in a separate work.

In this letter we offer a new way of thinking about the nature of phantom dark energy by revealing its connection to exotic crystalline structures that would normally be associated with matter rather than vacuum energy - stable time crystals. The latter (\(\rho>0, c_{s}^{2}> 0, g''(X)>0, w_{\phi } \le -1)\) satisfy the conditions required for the stability of perturbations8 for a phantom field15,16,17. Perhaps, similar relations with crystalline structures may be found in the future for other forms of dark energy, including a pure vacuum energy.

Note Added: As this paper was submitted for publication, the immensely exciting DESI collaboration results on dark energy were released18,19,20,21. When combined with other datasets they seem to hint at a time dependent dark energy that possibly crosses the phantom divide such that it recently loses power, dubbed \(w_{0}w_{a}CDM\) cosmology. From a phenomenological point of view, making models with a phantom divide crossing to accommodate the data can be achieved by a simple and specific change of behavior in the function \(g'(X)\) in the class discussed here, from negative to positive. Hence, such models would not be derived or founded in fundamental physics and would be hard to justify. Besides, the interpretation of data strongly depends on the parametrization of dark energy as a function of time. The parametrization inquiry and a micro physical understanding of dark energy after the phantom crossing remain open problems. Their further investigation would guide theorists to better motivated models that connect the two sides of the phantom divide.