Lorentzian Vacuum Transitions in $f(R)$ gravity

The Big Picture: The Universe's Great Escape

Imagine our universe is a ball sitting in a valley. In physics, this ball represents the "vacuum" (the state of empty space). Usually, the ball sits at the very bottom of the valley, which is the most stable, happy place. But sometimes, there's a "false vacuum"—a small dip in the hill where the ball is stuck, but it's not the lowest point possible.

Vacuum decay is the story of that ball suddenly realizing there is a deeper valley nearby and tunneling through the mountain to get there. This is a quantum event; the ball doesn't climb over the mountain, it magically appears on the other side.

This paper asks a big question: What happens to this "tunneling" process if the rules of gravity are slightly different from what Einstein taught us?

The Cast of Characters

The Old Rules (General Relativity/Einstein): The standard gravity we know. It says space-time is a flexible fabric, and mass tells it how to curve.
The New Rules (f(R) Gravity): The authors are testing a "modified" version of gravity. Imagine Einstein's gravity is a recipe for a cake. f(R) gravity is like adding a secret spice (a function of the curvature of space) to the recipe. It changes how the cake rises and bakes, but it's still a cake.
The Tunneling Ball (The Scalar Field): This is the thing trying to jump from the false vacuum to the true vacuum.
The Map (The Wheeler-DeWitt Equation): This is the "quantum map" the authors use to calculate the odds of the jump happening. It's a complex equation that describes the universe as a giant wave function.

The Method: A New Way to Look at the Map

Usually, physicists calculate these tunneling probabilities using a "Euclidean" approach. Think of this like looking at a map of a mountain range where time is turned into a fourth dimension of space. It's a bit like looking at a shadow of the mountain; it's useful, but you have to rotate the shadow back to reality to see what happens after the tunnel.

The authors use a Lorentzian approach. This is like looking at the mountain in real-time, in 3D, with time flowing normally. They don't have to rotate the map; they just watch the ball roll.

They also add Quantum Corrections.

Level 1 (Semiclassical): The ball is a solid object, but it has a tiny bit of "fuzziness" (quantum mechanics).
Level 2 (First Correction): The fuzziness gets a bit more detailed.
Level 3 (Second Correction): The fuzziness is fully accounted for.

The paper's main achievement is taking this "real-time" map and applying it to the "spiced" gravity recipes (f(R) theories) for the first time, going all the way up to the second level of quantum fuzziness.

The Experiments: Flat vs. Curved Universes

The authors ran simulations on two types of universes:

1. The Flat Universe (The Infinite Sheet)
Imagine the universe is an infinite, flat sheet of paper.

The Finding: When they added the "spice" (f(R) gravity), the probability of the universe tunneling changed. Sometimes the odds went up, sometimes down, depending on the specific spice used.
The Surprise: Even with the new gravity rules, the general behavior remained the same as Einstein's gravity. The universe still avoids a "singularity" (a point of infinite density, like the Big Bang starting from nothing) thanks to the quantum corrections. It's as if the quantum fuzziness acts like a safety net, preventing the ball from falling into an infinite hole.

2. The Closed Universe (The Balloon)
Imagine the universe is the surface of a balloon. It's finite but has no edges.

The Finding: Here, the "spice" caused a major conflict. In the old "shadow map" method (Euclidean), physicists assumed the curvature of the universe stayed constant during the jump. But in this "real-time" method, that assumption breaks down. The math says: "You can't have a constant curvature if the universe is expanding and changing shape at the same time."
The Result: They had to do some heavy approximation (guessing the shape of the curve in specific ranges), but even then, the result held: the universe still avoids the singularity, and the probability of the jump is generally lower than in the flat case.

The "Starobinsky" Model: The Star of the Show

One specific "spice" recipe is very famous: the Starobinsky model. It's a leading theory for how the universe expanded rapidly right after the Big Bang (Inflation).

The authors compared this model against a simple "square" gravity model and standard Einstein gravity.
Result: The Starobinsky model predicted the lowest probability for the universe to tunnel. It's the most "stubborn" universe, preferring to stay in its current state rather than jumping to a new one.

The Bottom Line: What Does This Mean?

Gravity is Robust: Even if we change the fundamental rules of gravity (f(R) theories), the universe behaves in a surprisingly similar way to Einstein's predictions. The "safety net" that prevents the universe from starting as a singularity seems to be a feature of quantum mechanics itself, not just a quirk of Einstein's specific gravity.
The "Spice" Matters: While the general behavior is the same, the exact odds of the universe tunneling depend heavily on which modified gravity theory is true. Some theories make the jump easier; others make it harder.
Time Matters: By using the "real-time" (Lorentzian) method instead of the "shadow" method, the authors found that some assumptions made in the past (like constant curvature in closed universes) might be wrong. This suggests our understanding of how the universe began might need a slight update.

In a nutshell: The authors took the complex math of quantum tunneling and applied it to new theories of gravity. They found that while the specific numbers change, the universe is surprisingly resilient. It seems that no matter how you tweak the laws of gravity, the quantum nature of the universe protects it from collapsing into a singularity, ensuring a "smooth" start to existence.

1. Problem Statement

The paper investigates the quantum mechanical probability of vacuum transitions (tunneling) between two minima of a scalar field potential within the framework of $f(R)$ modified gravity. While vacuum decay has been extensively studied in General Relativity (GR) using both Euclidean path integrals (Coleman-De Luccia) and Lorentzian Hamiltonian formalisms, the behavior of these transitions in modified gravity theories remains less explored.

Specifically, the authors aim to:

Extend the Lorentzian Hamiltonian approach (Wheeler-DeWitt equation) to $f(R)$ gravity, incorporating quantum corrections up to the second order in the semiclassical WKB expansion.
Determine how higher-order curvature terms in the gravitational action affect transition probabilities, the nucleation of universes, and the avoidance of initial singularities.
Compare results for spatially flat ( $k=0$ ) and closed ( $k=1$ ) Friedmann-Lemaître-Robertson-Walker (FLRW) universes.

2. Methodology

The authors employ a Lorentzian Hamiltonian formalism based on the canonical quantization of gravity, avoiding the Wick rotation required in Euclidean approaches.

Action and Variables: The system consists of an $f(R)$ gravity action coupled to a canonical scalar field $\phi$ with potential $V(\phi)$ . To handle the higher-order derivatives inherent in $f(R)$ , the Ricci scalar $R$ is treated as an independent dynamical variable (auxiliary field) via a Lagrange multiplier.
Hamiltonian Constraint: The authors derive the Hamiltonian constraint for minisuperspace models. For FLRW metrics, the minisuperspace variables are the scale factor $a$ , the Ricci scalar $R$ , and the scalar field $\phi$ . The constraint is quadratic in momenta:
$H = \frac{1}{2} G^{MN} \pi_M \pi_N + K[\Phi] \approx 0$
where $G^{MN}$ is the minisuperspace metric and $K$ contains the potential terms.
WKB Expansion: The wave function of the universe is expanded using the WKB ansatz $\Psi = \exp(iS/\hbar)$ $Ψ = exp (i S /ℏ)$ , where the action $S$ $S$ is expanded in powers of $\hbar$ $ℏ$ :
$S = S_0 + \hbar S_1 + \hbar^2 S_2 + \dots$
- $S_0$ : Classical action (semiclassical contribution).
- $S_1$ : First-order quantum correction.
- $S_2$ : Second-order quantum correction.
Transition Probability: The probability of tunneling from a false vacuum ( $\phi_A$ ) to a true vacuum ( $\phi_B$ ) is calculated as:
$P(A \to B) = \exp[-2 \text{Re}(\Gamma)]$
where $\Gamma = \Gamma_0 + \Gamma_1 + \Gamma_2$ represents the difference in the action between the two vacuum states, including the "wall" (bubble) contribution.
Approximations:
- Thin Wall Approximation: The bubble wall is treated as a thin interface.
- Constant Ricci Scalar: In specific sections, the authors assume $R$ is constant to simplify analytical calculations, a common assumption in Euclidean treatments.
- Power-Law Models: Explicit solutions are derived for $f(R) = R^{1+n}$ and the Starobinsky model ( $f(R) = R + \beta R^2$ ).

3. Key Contributions

Extension to Modified Gravity: The paper successfully generalizes the second-order WKB expansion of the Wheeler-DeWitt equation to $f(R)$ theories, providing a framework to compute transition probabilities beyond standard GR.
Lorentzian vs. Euclidean Discrepancy: A significant finding is that the assumption of a constant Ricci scalar, which is valid and motivated by symmetry in Euclidean approaches for closed universes, is inconsistent within the Lorentzian Hamiltonian formalism for closed FLRW metrics. This highlights a fundamental difference between the two formalisms.
Analytical Solutions for Non-Constant $R$ : The authors derive analytical expressions for transition probabilities without assuming a constant Ricci scalar for power-law models ( $R^{1+n}$ ) in the flat case, solving the coupled differential equations for $a$ and $R$ .
Higher-Order Corrections: The study explicitly calculates terms up to $\hbar^2$ , demonstrating that second-order corrections are crucial for avoiding initial singularities in these modified gravity scenarios.

4. Results

A. Flat FLRW Universe ( $k=0$ )

Constant Ricci Scalar: When assuming $R$ $R$ is constant, the general behavior of transition probabilities mirrors that of GR. The probabilities depend on the specific $f(R)$ $f (R)$ model via numerical factors and tension terms.
- Power-Law ( $R^{1+n}$ ): For $n < 2$ , the probability decays faster than in GR. For $n > 2$ , the decay is slower.
- Starobinsky Model ( $R + \beta R^2$ ): This model exhibits a faster probability decay compared to the pure quadratic $R^2$ model and GR.
Non-Constant Ricci Scalar: Without the constant $R$ $R$ assumption, the authors solved the system for $f(R) = R^{1+n}$ $f (R) = R^{1 + n}$ .
- The general behavior remains consistent with GR: the probability is bounded, and the specific value of $n$ determines whether the decay is faster or slower.
- Singularity Avoidance: Crucially, the inclusion of the second-order quantum correction ( $\Gamma_2$ ) leads to the avoidance of the initial singularity ( $a=0$ ). The probability distribution peaks at a finite, non-zero scale factor, similar to findings in GR but robust across different $f(R)$ models.
- Limit Behavior: The second-order correction term for power-law models does not smoothly reduce to the GR limit as $n \to 0$ , indicating a fundamental difference in the quantum structure of the minisuperspace at this order.

B. Closed FLRW Universe ( $k=1$ )

Inconsistency of Constant $R$ : The authors prove that a constant Ricci scalar is incompatible with the evolution of the scale factor in the Lorentzian Hamiltonian approach for closed universes.
Approximated Solutions: By analyzing regimes where $R$ $R$ does not diverge as $a \to 0$ $a \to 0$ , they derived approximate solutions.
- The results show that the transition probabilities for closed universes are consistently lower than those for flat universes.
- Despite the mathematical complexity and different functional dependencies, the general behavior is preserved: the second-order corrections still predict a non-singular initial state.

C. Specific Models

$f(R) = R^{1+n}$ : The transition probability is highly sensitive to the exponent $n$ . The tension terms (related to the bubble wall) acquire new dependencies on $n$ , sometimes changing sign or becoming complex for $n > 2$ .
Starobinsky Model: This model consistently yields the lowest transition probabilities among the cases studied, with a peak probability occurring at smaller scale factors compared to GR.

5. Significance

Robustness of Singularity Avoidance: The most significant physical result is that the mechanism for avoiding the initial singularity via second-order quantum corrections is robust. It persists not only in GR but also in a wide class of $f(R)$ modified gravity theories, regardless of whether the Ricci scalar is assumed constant or not.
Formalism Validation: The work validates the Lorentzian Hamiltonian approach as a viable tool for studying quantum cosmology in modified gravity, offering an alternative to Euclidean methods that avoids potential artifacts from analytic continuation.
Model Discrimination: The study provides specific predictions (e.g., decay rates, peak scale factors) that could theoretically distinguish between different $f(R)$ models and GR based on vacuum transition probabilities.
Theoretical Insight: The discovery of the inconsistency between the constant Ricci scalar assumption and the Lorentzian formalism in closed universes suggests that Euclidean and Lorentzian approaches may yield fundamentally different physical pictures in modified gravity, warranting further investigation.

In conclusion, the paper demonstrates that while $f(R)$ modifications alter the quantitative details of vacuum transition probabilities (making them model-dependent), the qualitative features of quantum cosmology—specifically the avoidance of the Big Bang singularity through higher-order quantum corrections—remain a stable feature of the theory.

Lorentzian Vacuum Transitions in f(R)f(R)f(R) gravity