Cosmology at the end of the world

TL;DR

This work demonstrates a holographic braneworld cosmology by embedding a 4D end-of-the-world brane in an AdS-Reissner-Nordström bulk and mapping the brane dynamics to a dual CFT state prepared via a Euclidean path integral. A positive Euclidean preparation time yields sensible geometries with the brane outside the horizon, and Lorentzian evolution yields a closed FLRW brane universe with a Big Bounce. Gravity localization on the brane is shown to be local and transient through long-lived tensor quasi-bound graviton modes, whose properties depend on brane tension, brane position, and bulk charge. These results provide a holographic framework for quantum cosmology and suggest possibilities for simulating cosmology on quantum hardware, while highlighting open issues on observables and stability.

Abstract

In the last two decades the Anti-de Sitter/Conformal Field Theory correspondence (AdS/CFT) has emerged as focal point of many research interests. In particular, it functions as a stepping stone to a still missing full quantum theory of gravity. In this context, a pivotal question is if and how cosmological physics can be studied using AdS/CFT. Motivated by string theory, braneworld cosmologies propose that our universe is a four-dimensional membrane embedded in a bulk five-dimensional AdS spacetime. We show how such a scenario can be microscopically realized in AdS/CFT using special field theory states dual to an "end-of-the-world brane" moving in a charged black hole spacetime. Observers on the brane experience cosmological physics and approximately four-dimensional gravity, at least locally in spacetime. This result opens a new path towards a description of quantum cosmology and the simulation of cosmology on quantum machines.

Figures (14)

• Figure 1: Brane trajectory. Maximally extended Penrose diagram for the AdS-Reissner-Nordström black hole. Only the time and radial directions are represented, while the other spatial dimensions are suppressed. Light rays move at an angle of $45^\circ$. More patches of the AdS-RN spacetime are glued together here. Region I is the exterior of the black hole and the CFT lives on the asymptotic boundary associated with it. Region I' is the corresponding second asymptotic region, typical of maximally extended black holes. In Regions II and IV, which are bounded by the outer and inner horizons, the radial coordinate is timelike, while the time coordinate is spacelike. Regions III and III' are the interior regions of the maximally extended black hole. The ETW brane oscillates inside and outside the two horizons of the black hole, cutting off the left asymptotic region I'. The red region, generally part of the maximally extended charged black hole, is not present in our spacetime.
• Figure 2: Euclidean brane trajectory. The Euclidean path integral on this spacetime computes the norm of the CFT state. The radial coordinate is the radius $r$, with the outer horizon $r=r_+$ at the center and the asymptotic boundary $r=\infty$ at the circumference. The other spatial dimensions are suppressed. The angular coordinate is the Euclidean time $\tau$, which has a periodicity $\beta$. The blue dashed line represents the $\tau=0,\pm\beta/2$ slice, where the brane reaches its minimum radius $r_0$. The red region is cut off by the ETW brane. The portion of asymptotic boundary not excised by the brane gives twice the preparation time $\tau_0$ of the CFT state.
• Figure 3: Euclidean sensible solutions. (a) Preparation time $\tau_0$ as a function of the ratio between inner and outer horizon radii $r_-/r_+$ for different values of the tension $T$. The larger the ratio, the closer the black hole to extremality. The preparation time is positive for every value of the tension if the black hole is sufficiently close to extremality. We chose four spatial dimensions ($d=4$), $r_+=100$ and we set the AdS radius to be $L_{AdS}=1$. (b) The ratio between the minimum radius of the brane $r_0$ and the outer horizon radius $r_+$ grows with the tension $T$. Our choice of parameters is $d=4$, $r_+=100$, $r_-=99.9$, $L_{AdS}=1$. When the brane is near-critical ($T\to 1/L_{AdS}$), it is far from the black hole horizon at the inversion point, and in the Lorentzian picture gravity can be locally localized on it.
• Figure 4: Potential $\mathbf{V_k[y(r^*)}$] - Large black hole. Potential (\ref{['potential']}) as a function of the tortoise coordinate $r^*$ in four spatial dimensions ($d=4$). We chose the values $r_+=100$, $r_-=99.9$ for the outer and inner horizon radii respectively, setting the AdS length to be $L_{AdS}=1$ and the angular momentum (and therefore the eigenvalue $k$) to be $l=k=1$. The potential diverges for $r^*_\infty=4.94\cdot 10^{-5}$ and vanishes exponentially at the horizon $r^*\to -\infty$.
• Figure 5: Quasi-bound mode. Results for four spatial dimensions ($d=4$), with outer and inner horizon radii given by $r_+=100$ and $r_-=99.9$ respectively. We set the AdS length to be $L_{AdS}=1$, giving $\gamma=L_{AdS}/r_+=0.01\ll 1$. The choice for the angular momentum is $l=5$, while the position of the brane is taken to be $y_b=34.62$ (corresponding to the maximum radius of a brane with tension $T=0.999999$). (a) Squared trapping coefficient $\xi(\omega)$ as a function of frequency $\omega$. $\xi(\omega)$ presents a Breit-Wigner peak centered at the real part $\bar{\omega}$ of the frequency of the quasi-bound mode, while its imaginary part $\Gamma/2$ is given by the half-width at half-maximum. The fit gives $\bar{\omega}=583.6$, $\Gamma=17.28$. The adjusted coefficient of determination for the fit is $R^2=0.999978$, showing the accuracy of the Breit-Wigner approximation for the peak of the squared trapping coefficient for this choice of parameters. (b) Quasi-bound mode wavefunction $\psi(r^*)$ as a function of the tortoise coordinate $r^*$. $\psi(r^*)$ oscillates in the near-horizon region, as shown in the top-left inset, and grows very fast near $r^*_b=4.65\cdot 10^{-5}$, as it is evident from the right-bottom inset. Therefore, the quasi-bound mode is very well localized on the brane, but has a non-vanishing probability to leak into the bulk and fall into the black hole.