What Happens Due to the Baby Universe Effect in Jackiw–Teitelboim Gravity? Analysis of Correlation Functions and Einstein–Rosen Bridge Length at Late Time Using Three Approaches

Abstract We analyze the correlation function in Jackiw–Teitelboim gravity using three approaches: by summing over all geodesics connecting boundary operators, integrating over the region of moduli space determined by the “no-shortcut condition,” and using the formula for the universal spectral density correlation in the $\tau$-scaling limit. We find that the behaviors of the three results coincide at late times: they all exhibit a “ramp” instead of permanent decay. Using the third approach we also confirm that the “plateau” appears after $T_H=2\pi e^{S_0}\widehat{\rho }_0(E)$. Overall, our results are consistent with the spectral form factor analysis. We also calculate the Einstein–Rosen bridge length $\langle \ell (T) \rangle$ using the three approaches and find that the results are in good agreement with each other. We also find that the $\langle \ell (T) \rangle$ grows as a cubic function in $T$ due to the contribution from geometry including one observable baby universe, and converges to a constant after $T=T_H$. For the geometry with one baby universe, we compute the size $\langle b(T) \rangle$ of the baby universe and find that it is of the same order as $\langle \ell (T) \rangle$. This result is consistent with the baby universe emission mechanism previously claimed.