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, \(\rho\).
