Does a wormhole survive a cosmological bounce?

This paper demonstrates that specific dynamical wormhole solutions in Einstein's equations can persist through a cosmological bounce without undergoing topology change, remaining valid across contracting, expanding, and bouncing phases of the universe.

The Gist

Imagine the history of our universe not as a single explosion starting from nothing (the Big Bang), but as a giant cosmic rubber band. It was stretched out, then it snapped back and contracted until it got incredibly tight, and then—boing!—it bounced back out again into the expansion we see today. This is called a "Cosmological Bounce."

Now, imagine there's a magical tunnel in space, a wormhole, that connects two distant points like a shortcut through a mountain. The big question this paper asks is: If the universe contracts, bounces, and expands again, does that wormhole survive the trip, or does it get crushed like a soda can?

Here is the breakdown of the scientists' findings, explained simply:

1. The Setup: The Cosmic Squeeze

In the standard "Big Bang" story, the universe started as a single, infinitely small point. But in this "Bouncing" story, the universe was already huge and full of stuff before it started shrinking.

• The Problem: When things get squeezed that hard, physics gets weird. Usually, we expect any complex structures (like black holes or wormholes) to be destroyed during the "squeeze" phase before the bounce.
• The Twist: Scientists already knew that black holes could survive this squeeze. They are tough. But what about wormholes?

2. The Two Contenders: Kim vs. Perez-Raia Neto

The researchers looked at two specific mathematical models of wormholes that exist inside an expanding/contracting universe. Think of these as two different blueprints for a cosmic tunnel.

Contender A: The Kim Wormhole (The "Fragile" One)

• How it works: This wormhole is like a tunnel with a specific, fixed size.
• The Result: It can survive the bounce, but only if it's small enough.
• The Catch: The paper calculates that if the wormhole is too "heavy" (larger than about 22 times the mass of our Sun), the cosmic squeeze will crush it. It's like trying to push a large boulder through a narrow tunnel; if the tunnel gets too tight, the boulder gets stuck or breaks the tunnel.
• Verdict: Small wormholes survive; giant ones might not.

Contender B: The Perez-Raia Neto Wormhole (The "Chameleon")

• How it works: This wormhole is special because it's made of "smart" material that changes size along with the universe. As the universe shrinks, the wormhole shrinks. As the universe expands, the wormhole expands.
• The Result: It survives perfectly, no matter how big or small it is.
• The Analogy: Imagine a rubber band tunnel. If you stretch the whole room, the rubber band stretches with it. If you squeeze the room, the rubber band shrinks. It never breaks because it adapts to the pressure.
• Verdict: This one is a survivor. It stays open before, during, and after the bounce.

3. The "Exotic" Fuel

To keep a wormhole open, you need something weird called "Exotic Matter."

• Normal Matter: Think of a rock or a star. It pulls things together (gravity). If you try to build a tunnel with normal matter, gravity will crush the tunnel shut.
• Exotic Matter: This is the "anti-gravity" fuel. It pushes things apart, holding the tunnel open against the crushing weight of the universe.
• The Finding: The paper confirms that the Perez-Raia Neto wormhole uses this exotic fuel. Even at the moment of the bounce (when the universe is most squeezed), this fuel keeps the tunnel open. It violates the usual rules of physics, but that's exactly what wormholes need to do.

4. What Happens at the Bounce?

At the exact moment of the bounce, the universe stops shrinking and starts expanding.

• For the Kim wormhole, the throat (the narrowest part of the tunnel) gets very small but doesn't close completely, provided it wasn't too big to begin with.
• For the Perez-Raia Neto wormhole, the throat gets smaller as the universe shrinks, reaches a minimum size at the bounce, and then grows again as the universe expands. It never closes.

5. Why Should We Care? (The "What If" Scenarios)

If these wormholes survived the bounce, they could be hiding in our universe right now!

• Cosmic Jets: Imagine gas falling into these ancient wormholes. It could shoot out like a powerful jet, helping to light up the early universe and form the first stars.
• Time Travel (Sort of): They could act as a bridge between the "before" universe (the contracting phase) and our current "after" universe.
• Gravitational Waves: If these tunnels exist, they might be sending ripples through space (gravitational waves) that we could detect with our telescopes.

The Bottom Line

The universe might have been squeezed like a sponge, but the paper shows that wormholes are tough enough to survive the squeeze.

• Some wormholes (the Kim type) need to be small to make it.
• Others (the Perez-Raia Neto type) are so adaptable that they survive no matter what.

So, the next time you look at the stars, remember: somewhere out there, a cosmic tunnel might have been there since before the Big Bang, surviving the entire history of the universe to connect two points in space today.

Technical Summary

1. Problem Statement

The standard $\Lambda$-CDM cosmological model posits an initial singularity (the Big Bang) where general relativity breaks down. To resolve this, bouncing cosmologies have been proposed, where the universe undergoes a contraction phase followed by a bounce into the current expansion phase, avoiding the singularity.

A critical question in this context is the fate of topological structures formed during the contraction phase. While previous studies have shown that black holes can survive a bounce, the survival of wormholes (regions of spacetime with non-trivial topology) remains an open problem. Unlike black holes, wormholes require "exotic matter" violating the Null Energy Condition (NEC) to remain open. The authors investigate whether dynamical wormholes embedded in a cosmological background can persist through the transition from a contracting to an expanding universe without undergoing a topology change (i.e., closing up).

2. Methodology

The authors employ a rigorous geometric and causal analysis within the framework of General Relativity.

• Models Analyzed: They focus on the only two known exact solutions of Einstein's equations representing dynamical wormholes in a cosmological (FLRW) background:
  1. The Kim solution [34–36].
  2. The Pérez-Raia Neto (PRN) solution [37].
• Bouncing Background: They utilize a phenomenological scale factor $a(t)$ to model the bounce:
  $$a(t) = a_b \left[ 1 + \left(\frac{t}{T_b}\right)^2 \right]^{1/3}$$
  This model ensures a non-singular bounce at $t=0$ (where $\dot{a}=0$) and asymptotically approaches a dust-dominated universe at late times.
• Throat Definition: Instead of the static Morris-Thorne definition, they adopt the dynamic definition by Hochberg and Visser. The throat is defined as a minimal 2-surface where the expansion of null geodesics ($\theta$) vanishes ($\theta=0$) and satisfies a generalized flare-out condition ($n^a \nabla_a \theta \geq 0$).
• Survival Criterion: A wormhole survives the bounce if the throat definition and the flare-out condition hold for all cosmic times ($t < 0$, $t=0$, and $t > 0$). If the throat closes or the flare-out condition fails, a topology change occurs.
• Matter Analysis: They calculate the energy-momentum tensor components (density, pressures, and heat flux) to verify NEC violation and analyze the causal structure via null geodesics and embedding diagrams.

3. Key Contributions and Results

A. The Kim Cosmological Wormhole

• Geometry: The metric is a conformal scaling of the Morris-Thorne wormhole.
• Throat Location: The areal radius of the throat $R_-$ is time-dependent.
• Survival Condition: The authors find that the wormhole only survives if the throat parameter $b_0$ (associated with mass $M$) satisfies a strict upper bound:
  $$b_0 \leq \frac{GM^*}{c^2}, \quad \text{where } M^* \approx 22 M_\odot$$
• Failure Mode: If $b_0 > 22 M_\odot$, the condition $1 - 4b_0^2 H^2 a^2(t) > 0$ (required for the throat to exist) is violated at certain times, particularly near the bounce. Consequently, the flare-out condition fails, and the wormhole closes or becomes undefined.
• Conclusion: Survival is conditional on the mass of the wormhole.

B. The Pérez-Raia Neto (PRN) Cosmological Wormhole

• Geometry: This solution involves an imperfect fluid with heat flux. The throat location in isotropic coordinates is $\tilde{r}_{th} = b_0 / (2a(t))$.
• Survival Condition: The authors demonstrate that the throat is perfectly defined for all cosmic times, regardless of the value of $b_0$.
• Flare-out Condition: The generalized flare-out condition is satisfied at all times, including the bounce ($t=0$).
• Causal Structure: Analysis of null geodesics confirms that light cones remain well-behaved, and no geodesics cross the throat from the "forbidden" region ($R < b_0$).
• Conclusion: The PRN wormhole unconditionally survives the cosmological bounce.

C. Matter Content and Energy Conditions

• NEC Violation: For the PRN solution, the authors explicitly prove that the Null Energy Condition is violated at the throat for all times ($t < 0, 0, > 0$). This is a necessary condition for the wormhole to remain open.
• Heat Flux: The PRN solution features a non-zero heat flux $q$. The analysis shows that:
  • Before the bounce ($t < 0$), heat flows toward the throat.
  • After the bounce ($t > 0$), heat flows away from the throat.
  • The flux vanishes rapidly as $t \to \infty$.
• Embedding Diagrams: The embedding diagrams show that while the physical size of the wormhole scales with the universe (contracting before the bounce, expanding after), the topological "shape" (the throat) remains intact.

4. Significance and Implications

• Topological Stability: The paper provides the first rigorous proof that specific dynamical wormhole solutions can survive a non-singular cosmological bounce without undergoing topology change. This challenges the assumption that all pre-bounce structures must be destroyed.
• Astrophysical Manifestations:
  • Early Universe Jets: Surviving wormholes could accrete gas in the early universe, producing relativistic jets that contribute to re-ionization and the formation of the first structures.
  • Cosmic Nozzles: Wormholes connecting the contracting and expanding phases could act as "Laval nozzles," converting thermal energy into kinetic energy, potentially driving supersonic winds that influence early star formation.
  • Gravitational Waves: These objects could act as channels for gravitational waves, potentially leaving distinct signatures in the stochastic gravitational wave background.
• Model Dependence: The results highlight that survival is model-dependent. While the PRN solution is robust, the Kim solution imposes strict mass limits. This suggests that not all wormhole geometries are compatible with bouncing cosmologies.

5. Conclusion

The authors conclude that in a bouncing universe, dynamical wormholes can indeed survive the transition from contraction to expansion. Specifically, the Pérez-Raia Neto solution survives unconditionally, while the Kim solution survives only for wormholes with masses below $\approx 22 M_\odot$. This implies that if the universe underwent a bounce, topological defects like wormholes could persist into the current expanding epoch, offering new avenues for astrophysical observation and cosmological theory.