Fate of the False Vacuum: Finite Temperature, Entropy, and Topological Phase in Quantum Simulations of the Early Universe

Despite being at the heart of the theory of the “Big Bang” and cosmic inflation, the quantum-field-theory prediction of false vacuum tunneling has not been tested. To address the exponential complexity of the problem, a table-top quantum simulator in the form of an engineered Bose-Einstein condensate (BEC) has been proposed to give dynamical solutions of the quantum-field equations. In this paper, we give a numerical feasibility study of the BEC quantum simulator under realistic conditions and temperatures, with an approximate truncated Wigner phase-space method. We report the observation of false vacuum tunneling in these simulations, and the formation of multiple bubble “universes” with distinct topological properties. The tunneling gives a transition of the relative phase of coupled Bose fields from a metastable to a stable “vacuum.” We include finite-temperature effects that would be found in a laboratory experiment and also analyze the cutoff dependence of modulational instabilities in Floquet space. Our numerical phase-space model does not use thin-wall approximations, which are inapplicable to cosmologically interesting models. It is expected to give the correct quantum treatment, including superpositions and entanglement during dynamics. By analyzing a nonlocal observable called the topological phase entropy (TPE), our simulations provide information about phase structure in the true vacuum. We observe a cooperative effect in which true vacua bubbles representing distinct universes each have one or the other of two distinct topologies. The TPE initially increases with time, reaching a peak as multiple universes are formed, and then decreases with time to the phase-ordered vacuum state. This gives a model for the formation of universes with one of two distinct phases, which is a possible solution to the problem of particle-antiparticle asymmetry.

Popular Summary

The universe is not a Clockwork Orange, despite the popularity of the movie. Instead, it is thought that quantum effects may have dominated the very early universe, causing density fluctuations in the cosmic microwave background. However, even though at the heart of modern cosmology, the quantum prediction of false vacuum tunneling leading to a “Big Bang” has not been tested. We give a numerical feasibility study of a table-top Bose-Einstein condensate quantum-simulator proposal for this effect under realistic conditions. This shows the viability of an experimental laboratory demonstration.

We report false vacuum tunneling in computer simulations, and the formation of multiple bubble universes with distinct topological properties. The tunneling gives a transition of the relative phase of coupled Bose fields from a metastable to a stable vacuum. The finite-temperature effects of a laboratory experiment are included. We also analyze modulational instabilities in Floquet space, using an approximate truncated Wigner method.

We define a nonlocal observable called the topological phase entropy. A cooperative effect occurs, in which the true vacua bubbles representing distinct universes each have one or the other of two distinct topologies. The entropy initially increases with time, reaching a peak as multiple universes are formed, and then decreases with time to the phase-ordered vacuum state. This gives a model for the formation of universes with one of two distinct phases, which is a possible solution to the problem of particle-antiparticle asymmetry.

Figure 1

The engineered potential for Eq. (9), with $\lambda =1.0,1.2,1.5$, showing a local minimum for $\lambda >1$. Here ${\omega }_0=1$ for purposes of illustration.

Figure 2

Example of a single trajectory decay to a true vacuum, starting in a 1D false vacuum initialized with finite-temperature effects. The false vacuum is seen to decay to two distinct topological phases ${\varphi }_a(\tilde{x})$ for the true vacuum, indicated by yellow and light-blue regions. The bubbles of true vacuum expand in dimensionless space $\tilde{x}$ until they meet each other. At dimensionless long time scales $\tilde{t}$, the universes are separated by domain walls of false vacuum, indicated by the color green. Dimensionless parameters are $\tau ={10}^{-5}$, $\tilde{\nu }=7\times {10}^{-3}$, $\lambda =1.2$, $\tilde{\omega }=50$, and $\tilde{\rho }=200$. The definition of the parameters is given in Sec. 3 5c.

Figure 3

Single-trajectory 1D false vacuum simulation for the time evolution of $p_z$ at $\tau =1\times {10}^{-5}$. The dimensionless length $\tilde{L}=100$ corresponds to a trap circumference $L=254\mu \mathrm{m}$. Simulation parameters are $\lambda =1.2$, $\tilde{\nu }=7\times {10}^{-3}$, $\tilde{\omega }=50$, $\tilde{\rho }=200$, number of modes $M=256$. The false vacuum ($p_z=1$) indicated by the yellow color decays, forming bubbles in the true vacua ($p_z=-1$) indicated by the blue regions.

Figure 4

Single-trajectory 1D false vacuum simulation for the time evolution of $p_z$ at $\tau =3\times {10}^{-4}$, all other parameters are as in Fig. 3.

Figure 5

Time evolution of the corresponding average relative phase $⟨\cos {\varphi }_a⟩$ shown in Fig. 3 (solid line) and Fig. 4 (dash line), the horizontal dash-dot line $⟨\cos {\varphi }_a⟩=0.9$ indicates the threshold of the appearance of a true vacuum bubble.

Figure 6

Dependence of the tunneling rate $\mathrm{\Gamma }$ on the coupling $\tilde{\nu }$ for different values of reduced temperature $\tau$ for $\lambda =1.2$. The error bars show the estimated error of the linear least-squares fitting in log scale.

Figure 7

Tunneling rate $\mathrm{\Gamma }$ for different values of reduced temperature $\tau$ for $\lambda =1.3$.

Figure 8

Tunneling rate $\mathrm{\Gamma }$ for different values of reduced temperature $\tau$ for $\lambda =1.4$.

Figure 9

Evolution of the topological phase entropy function $S$ in early universe simulations using ${10}^4$ trajectories with varying order of magnitude, of reduced temperature $\tau$. Here the selected dimensionless parameters are $\tilde{\nu }=7\times {10}^{-3}$, $\lambda =1.2$, $\tilde{\omega }=50$, and $\tilde{\rho }=200$ with simulation parameters $\tilde{L}=100$, space-phase bins $\ell =8$, $p=3$, with $N_{\mathrm{grid}}=300$ and $8\times {10}^4$ time steps.

Figure 10

(a) Dependence of the entropy function $S(\tilde{t})$ on coupling $\tilde{\nu }$ calculated using ${10}^4$ trajectories with $\lambda =1.3$, $\tilde{\omega }=50$, $\tilde{\rho }=200$, and $\tau ={10}^{-5}$. (b) $⟨\cos ({\varphi }_a)⟩=(1/\tilde{L})\underset{0}{\overset{\tilde{L}}{\int }}\cos {\varphi }_a(\tilde{x})d\tilde{x}$ for single-trajectory simulations for different values of $\tilde{\nu }$. The horizontal dash-dot line $⟨\cos ({\varphi }_a)⟩=0.9$ indicates the threshold of the appearance of a true vacuum bubble. The tunneling time is longer for larger couplings $\tilde{\nu }$.

Figure 11

(a) Time evolution of the entropy $S(\tilde{t})$ using ${10}^4$ trajectories for the number of modes $M=300$, $512$, and $1024$. The dimensionless parameters chosen here are $\tilde{\omega }=50$, $\tau ={10}^{-5}$, $\tilde{\nu }=7\times {10}^{-3}$, $\lambda =1.2$, and $\tilde{\rho }=200$. (b) Time evolution of $⟨\cos ({\varphi }_a)⟩$ for different numbers of modes $M$. The horizontal dash-dotted line indicates a uniform random phase.

Figure 12

Critical wave number ${\tilde{k}}_c$ and Nyquist wave number ${\tilde{k}}_{\mathrm{Nyq}}=\pi /\mathrm{\Delta }\tilde{x}$ in the simulations for $\tilde{\omega }=50$ at various values of $\tilde{\nu }$ with fixed length $\tilde{L}=100$. For $M=256$, ${\tilde{k}}_{\mathrm{Nyq}}$ in the simulations are all below ${\tilde{k}}_c$, hence the unstable modes are excluded. For $M=1024$, the unstable modes are included, as ${\tilde{k}}_{\mathrm{Nyq}}>{\tilde{k}}_c$.

Figure 13

Single-trajectory 1D false vacuum simulation for the time evolution of $p_z$ at $\tau =1\times {10}^{-4}$ with $M=1024$ to include the effect of Floquet modes. The modulation frequency is $\tilde{\omega }=50$. Dimensionless parameters are $\tilde{L}=100$, $\lambda =1.2$, $\tilde{\nu }=7\times {10}^{-3}$, $\tilde{\rho }=200$. Bubbles in the true vacua ($p_z=-1$) are short lived and dominated by fluctuations at later time.

Figure 14

Time evolution of average relative phase $⟨\cos {\varphi }_a⟩$ using $8000$ trajectories for $M=256$ (${\tilde{k}}_{\mathrm{Nyq}}\approx 8.01<{\tilde{k}}_c\approx 15.39$, Floquet modes excluded) and $M=1024$ (${\tilde{k}}_{\mathrm{Nyq}}\approx 32.14>{\tilde{k}}_c\approx 15.39$, Floquet modes included). All other parameters are as in Fig. 13. The errors in $⟨\cos {\varphi }_a⟩$ are less than $1\mathrm{\%}$.

Figure 15

Time evolution of average relative phase $⟨\cos {\varphi }_a⟩$ using $8000$ trajectories for $\lambda =1.1$ (solid line), $\lambda =1.3$ (dash line), and $\lambda =1.5$ (dot line). Floquet modes are included by setting number of modes $M=1024$ (${\tilde{k}}_{\mathrm{Nyq}}\approx 32.14>{\tilde{k}}_c\approx 15.39$). Reduced temperature $\tau =1\times {10}^{-4}$, other dimensionless parameters $\tilde{L}=100$, $\tilde{\omega }=50$, $\tilde{\nu }=7\times {10}^{-3}$, $\tilde{\rho }=200$. The errors in $⟨\cos {\varphi }_a⟩$ are less than $1\mathrm{\%}$.

Figure 16

Single-trajectory simulation for $p_z$ at $\tau =1\times {10}^{-4}$ with $M=1024$. The modulation frequency is $\tilde{\omega }=150$. Floquet modes are included: ${\tilde{k}}_{\mathrm{Nyq}}\approx 32.1>{\tilde{k}}_c\approx 28.8$. Dimensionless parameters $\tilde{L}=100$, $\lambda =1.2$, $\tilde{\nu }=7\times {10}^{-3}$, $\tilde{\rho }=200$. Comparing to Fig. 13, the true vacuum ($p_z=-1$) is relatively long lived and stable.

Figure 17

Single-trajectory simulation for the time evolution of $p_z$ with an increased frequency $\tilde{\omega }=200$ to remove the Floquet modes (${\tilde{k}}_{\mathrm{Nyq}}\approx 32.14<{\tilde{k}}_c\approx 33.57$). Other parameters are as in Fig. 16.

Figure 18

Time evolution of $⟨\cos {\varphi }_a⟩$ using $8000$ trajectories for $\tilde{\omega }=50$ (solid line), $150$ (dash line), and $200$ (dot line), all other parameters are as in Fig. 16. The errors in $⟨\cos {\varphi }_a⟩$ are less than $1\mathrm{\%}$.