Wormhole spacetimes in an expanding universe: energy conditions and future singularities

TL;DR

The paper develops a four-scalar non-linear sigma model in which the target-space metric is identified with the spacetime Ricci tensor, enabling stable wormhole geometries even when traditional energy conditions are violated. By embedding these wormholes into an expanding FLRW universe, the authors show that the cosmological fluid can modify the total energy density and pressure, potentially satisfying the energy conditions in certain regimes, particularly when the wormhole throat is comparable to or larger than the Hubble radius. They also present scenarios where our universe encounters a finite future singularity while a connected universe via the wormhole remains nonsingular, with the throat becoming timelike near the singular surface to allow traversal. In addition, the work explores DM-halo motivated wormholes, demonstrating that nonzero pressures arise naturally in this framework and that such geometries can be realized without real dark matter, thanks to the four-scalar model. Overall, the results suggest rich wormhole phenomenology in cosmological and astrophysical settings, with potential implications for topology, stability, and observational signatures.

Abstract

We study wormhole geometries embedded in an expanding universe within a four-scalar non-linear $σ$ model, where the target-space metric is identified with the spacetime Ricci tensor. In this framework, wormholes can remain stable even when conventional energy conditions are violated. However, once cosmological expansion is included, the effective energy density and pressure are modified by the cosmological fluid, enabling the energy conditions to be satisfied. We further present intriguing geometries in which a finite future singularity appears in our universe but not in another universe connected by the wormhole. Near the throat, the hypersurface becomes timelike, allowing trajectories to traverse to the other universe before the singularity and return afterwards. We also construct wormhole solutions motivated by galactic dark-matter halo profiles, where the required non-vanishing pressure arises naturally from the four-scalar non-linear $σ$ model.

Figures (7)

• Figure 1: The behaviours of $\rho$\ref{['fig1:sub1']}, $\rho + p^\mathrm{radial}$\ref{['fig1:sub2']}, $\rho + p^\mathrm{angular}$\ref{['fig1:sub3']}, $\rho + p^\mathrm{radial} + 2 p^\mathrm{angular}$\ref{['fig1:sub4']}, $\rho - \left| p^\mathrm{radial}\right|$\ref{['fig1:sub5']} and $\rho - \left| p^\mathrm{angular} \right|$\ref{['fig1:sub6']}, when we use the normalisation of $\frac{{H_0}^2}{\kappa^2}$ with $r_0 H_0 = 10^{-2}$, $w=-1/3$. The red plane indicates $z=0$, and each quantity takes positive values above this plane. All the energy conditions are violated, but they all might be satisfied in the late time.
• Figure 2: The behaviours of $\rho$\ref{['fig3:sub1']}, $\rho + p^\mathrm{radial}$\ref{['fig3:sub2']}, $\rho + p^\mathrm{angular}$\ref{['fig3:sub3']}, $\rho + p^\mathrm{radial} + 2 p^\mathrm{angular}$\ref{['fig3:sub4']}, $\rho - \left| p^\mathrm{radial}\right|$\ref{['fig3:sub5']} and $\rho - \left| p^\mathrm{angular} \right|$\ref{['fig3:sub6']}, when we use the normalisation of $\frac{{H_0}^2}{\kappa^2}$ with $r_0 H_0 = 10^{0}$, $w=-1/3$. The red plane indicates $z=0$, and each quantity takes positive values above this plane. All the energy conditions do not seem to be violated.
• Figure 3: The behaviours of $\rho$\ref{['fig2:sub1']}, $\rho + p^\mathrm{radial}$\ref{['fig2:sub2']}, $\rho + p^\mathrm{angular}$\ref{['fig2:sub3']}, $\rho + p^\mathrm{radial} + 2 p^\mathrm{angular}$\ref{['fig2:sub4']}, $\rho - \left| p^\mathrm{radial}\right|$\ref{['fig2:sub5']} and $\rho - \left| p^\mathrm{angular} \right|$\ref{['fig2:sub6']}, when we use the normalisation of $\frac{{H_0}^2}{\kappa^2}$ with $r_0 H_0 = 10^{-2}$, $w=-2/3$. All the energy conditions are violated, but they all might be recovered in the late time, except SEC, which is always violated.
• Figure 4: The behaviours of $\rho$\ref{['fig4:sub1']}, $\rho + p^\mathrm{radial}$\ref{['fig4:sub2']}, $\rho + p^\mathrm{angular}$\ref{['fig4:sub3']}, $\rho + p^\mathrm{radial} + 2 p^\mathrm{angular}$\ref{['fig4:sub4']}, $\rho - \left| p^\mathrm{radial}\right|$\ref{['fig4:sub5']} and $\rho - \left| p^\mathrm{angular} \right|$\ref{['fig4:sub6']}, when we use the normalisation of $\frac{{H_0}^2}{\kappa^2}$ with $r_0 H_0 = 10^{0}$, $w=-2/3$. The red plane indicates $z=0$, and each quantity takes positive values above this plane. All the energy conditions are violated only in the early time, although the SEC is always violated.
• Figure 5: The behaviours of $\rho$\ref{['fig5:sub1']}, $\rho + p^\mathrm{radial}$\ref{['fig5:sub2']}, $\rho + p^\mathrm{angular}$\ref{['fig5:sub3']}, $\rho + p^\mathrm{radial} + 2 p^\mathrm{angular}$\ref{['fig5:sub4']}, $\rho - \left| p^\mathrm{radial}\right|$\ref{['fig5:sub5']} and $\rho - \left| p^\mathrm{angular} \right|$\ref{['fig5:sub6']}, when we use the normalisation of $\frac{{H_0}^2}{\kappa^2}$ with $r_0 H_0 = 10^{-2}$, $w=-1$. The red plane indicates $z=0$, and each quantity takes positive values above this plane. Although all the energy conditions are violated, they are all recovered in the late time, except SEC, which is always violated.