Holographic (de)confinement transitions in cosmological backgrounds

For type IIB supergravity with a running axio-dilaton, we construct bulk solutions which admit a cosmological background metric of Friedmann-Robertson-Walker type. These solutions include both a dark radiation term in the bulk as well as a four-dimensional (boundary) cosmological constant, while gravity at the boundary remains nondynamical. We holographically calculate the stress-energy tensor, showing that it consists of two contributions: The first one, generated by the dark radiation term, leads to the thermal fluid of $\mathcal{N}=4$ SYM theory, while the second, the conformal anomaly, originates from the boundary cosmological constant. Conservation of the boundary stress-tensor implies that the boundary cosmological constant is time-independent, such that there is no exchange between the two stress-tensor contributions. We then study (de)confinement by evaluating the Wilson loop in these backgrounds. While the dark radiation term favors deconfinement, a negative cosmological constant drives the system into a confined phase. When both contributions are present, we find an oscillating universe with negative cosmological constant which undergoes periodic (de)confinement transitions as the scale of three-space expands and recontracts.

Figure 1

Plots of $e^{\varphi }$ vs $r$ for (a) $\lambda =-1-4{\mu }^2{\tilde{c}}_0^{1/2}$, (b) $\lambda =-4{\mu }^2{\tilde{c}}_0^{1/2}$ and (c) $\lambda =1-4{\mu }^2{\tilde{c}}_0^{1/2}$. Cases (a) and (c) are taken as examples for $\lambda <-4{\mu }^2{\tilde{c}}_0^{1/2}$ and $\lambda >-4{\mu }^2{\tilde{c}}_0^{1/2}$, respectively. Other parameters are set as $1/\mu =R=1$, $q=2$, and ${\tilde{c}}_0=0.1$. In the case of (c), $e^{\varphi }$ diverges at the horizon $r_H=R\sqrt{{\tilde{c}}_0^{1/2}+\frac{\lambda }{4{\mu }^2}}$, which is 0.5 in this case. Note that in general, ${\tilde{c}}_0$ is explicitly time-dependent, these curves represent snapshots of the dilaton solution at constant time.

Figure 2

Plot of $e^{\varphi (0)}$ vs $|\lambda |$, where the parameters are set as in Fig. 1, yielding $4{\mu }^2{\tilde{c}}_0^{1/2}=1.265$. We notice that $e^{\varphi (0)}$ is real for $|\lambda |>4{\mu }^2{\tilde{c}}_0^{1/2}=1.265$, which corresponds to case (a) in Fig. 1, with no horizon present. For $|\lambda |<4{\mu }^2{\tilde{c}}_0^{1/2}=1.265$, case (c) of Fig. 1, the dilaton becomes complex in the region hidden behind the horizon. Moreover, $e^{\varphi (0)}\to 1$ for $|\lambda |\to \infty$. Note that since in general ${\tilde{c}}_0$ is explicitly time-dependent, this curve should be understood at a given instance in time.

Figure 3

Plots of $n$ vs $r$ for (a) $\lambda =-1-4{\mu }^2{\tilde{c}}_0^{1/2}$, (b) $\lambda =-4{\mu }^2{\tilde{c}}_0^{1/2}$ and (c) $\lambda =1-4{\mu }^2{\tilde{c}}_0^{1/2}$. Cases (a) and (c) are taken as examples for $\lambda <-4{\mu }^2{\tilde{c}}_0^{1/2}$ and $\lambda >-4{\mu }^2{\tilde{c}}_0^{1/2}$, respectively. In agreement with (63), (a) and (b) show minima and hence confine. The parameters are taken to be $1/\mu =R=1$, $k=-1$, and ${\tilde{c}}_0=0.2$. For (c), there is a horizon at $r=0.5$. Note that for the parameter values chosen in this plot, $\lambda <0$ in all cases and hence $k=-1$ is consistent when solving the Friedmann Eq. (14). $k$ only enters the dilaton and hence the Wilson loop via the value of ${\tilde{c}}_0$. Furthermore, since in general ${\tilde{c}}_0$ is explicitly time-dependent, these curves represent snapshots at constant times.

Figure 4

Plots of $E$ vs $L$ for (A) $\lambda =-1-4{\mu }^2{\tilde{c}}_0^{1/2}$ and (B) $\lambda =+1-4{\mu }^2{\tilde{c}}_0^{1/2}$. Cases (A) and (B) are taken as examples for $\lambda <-4{\mu }^2{\tilde{c}}_0^{1/2}$ and $\lambda >-4{\mu }^2{\tilde{c}}_0^{1/2}$, respectively. Case (A) shows confinement, while case (B) shows typical screening behavior: In the latter case there exist two U-shaped configurations for small $L$, while for large separations the string breaks at the horizon and hence has vanishing energy (at $O(N^2)$). Other parameters are set as $q=0$, $\mu =1$, $R=1$, $k=-1$, ${\tilde{c}}_0=1.0$, $r_{\max }=3$ and ${\alpha }^{\prime }=1$. Note that in general ${\tilde{c}}_0$ is explicitly time-dependent, these curves represent snapshots of the dilaton solution at constant time.

Figure 5

The (de)confinement transition as seen in the Wilson loop behavior in an oscillating, open, universe ($\lambda <0$, $k=-1$). During the periods depicted in blue, the Wilson loop shows confining behavior, while during the red periods (close to the big bang singularity), the plasma is in a deconfined phase. The transition line, shown in black, is fixed by the value of the dark radiation constant $C$.