A Menagerie of Wormholes and Cosmologies in the Gravitational Path Integral

Abstract

We analyse a variety of Euclidean saddles in the gravitational path integral, with asymptotic AdS boundary conditions, in a class of Einstein-Scalar-Maxwell models. These include single boundary solutions, usual and wineglass wormholes, as well as more exotic (quasi)-oscillatory saddles. We find several interesting phase transitions between these solutions. The Euclidean wormhole backgrounds can be analytically continued to Lorentzian FLRW universes. Some of them contain an early period of inflation. We delineate the conditions under which they can be the dominant saddles in the gravitational path integral and use them to estimate ratios of probabilities for different cosmological outcomes.

Figures (12)

• Figure 1: The different types of Euclidean AdS gravitational saddles that we analyze and compare in this work. The first two types lead to crunching cosmologies, while the last two to inflating/expanding cosmologies upon analytic continuation ($\tau = i t$).
• Figure 2: $H_{\rm UV}$ and $\pi_A$ as functions of $H_{0,C}$ for the connected wormhole, as given by Eq. \ref{['eq:Huv_connected']}. We observe that while for large source $H_{\rm UV}$ there is only a single branch of wormholes, for smaller $H_{\rm UV}$ there can exist two branches. There is always a minimum value of $H_{\rm UV}\, (\text{or}\,\pi_A)$ , namely $H_{\rm UV}^{\rm min} \simeq 2.687/(\kappa|\tilde{V}_{\rm AdS}|)^{1/2}$ ($\pi_A^{\rm min} \simeq 2.613/(\kappa|\tilde{V}_{\rm AdS}|)^{1/2}$) , at which the two wormholes merge into a single one. Below this value, no wormhole exists. This plot can also be extended to negative $H_{\rm UV} \,(\text{or}\,\pi_A)\,,\,H_{0,C}$ being an odd function with a discontinuity at zero.
• Figure 3: The scale factor (upper) and its first (lower left) and second (lower right) derivatives as functions of the Euclidean time $\tau$ for the EAdS-like (blue), and EdS-like (blue dashed) regions described in eq. \ref{['eq:background_ans']}, all in $\kappa=1$ units. We have made the choice of parameters $H_{\rm AdS} =1.1$, $H_{\rm dS} = 1$ and $A_{\rm dS}=1$. The parameter $A_{\rm AdS}$ is in general freely tunable; however, we plot here the case that $c_{\rm AdS}=-2 \sqrt{A_{\rm AdS}(A_{\rm AdS}-A_{\rm dS})} =-2$, that corresponds to a configuration with a vanishing source for the scalar field. The resulting background inflates upon rotating $\tau = i t$ and evolving in Lorentzian signature after $t = 0$.
• Figure 4: $\tilde{V}$ vs. $\tilde{\phi}$ (upper), $\tilde{V}$ vs. $\tau$ (lower left) and $\tilde{\phi}$ vs. $\tau$ (lower right), all in $\kappa=1$ units. The parameters used are the same with those in fig. \ref{['fig:scale_fac']}. The chosen field value at the EAdS boundary is $\tilde{\phi}(\tau\rightarrow-\infty)=0$.
• Figure 5: $H_{\rm UV}$ (solid lines), and $\pi_A$ (dashed lines) as a functions of $H_{0,WG}$, and $H_{0,n}$ as given by \ref{['eq:Huv_wineglass']} and \ref{['eq:Huv_wineglass_OSC']} for the wineglass (left) and the oscillatory wineglass (right) wormholes in the case of Dirichlet BCs for the scalar field ($A_{\rm AdS}=\frac{1+\sqrt{5}}{2}, A_{\rm dS}=1$). $(\kappa|\tilde{V}_{\rm AdS}|)^{1/2}(H_{\rm UV}-\pi_A) < 10^{-3}$ is very small and not evident in the scale of this plot. The plot is also restricted to values of $H_{0,WG}$ and $H_{0,n}$ that satisfy the bound in eq. \ref{['eq:H02_ineq']}. We observe that only a single branch of wormhole solutions exists. The relation is a power law for large $H_{0,WG/n}$ (see \ref{['eq:H_UV/H_0behaviorwineglass']}). The plot can be extended to negative $H_{\rm UV},\,H_{0,WG/n}(H_{\rm UV})$ being an odd function.