IR behaviour of one-loop complex $\mathbb{R}\times S^3$ saddles

This paper investigates the infrared behavior of one-loop complex $\mathbb{R}\times S^3$ saddles in 4D Lorentzian Einstein-Hilbert gravity by computing the renormalized Hartle-Hawking wavefunction under Dirichlet and fixed extrinsic curvature boundary conditions, revealing that metric fluctuations induce secularly growing infrared divergences similar to those in pure Lorentzian de Sitter space, while confirming that all considered saddles remain KSW-allowed.

The Gist

Imagine the universe not as a static stage, but as a giant, breathing entity that is constantly trying to figure out its own shape. Physicists use a mathematical tool called a Path Integral to calculate the "wave function" of the universe. Think of this wave function as a probability map that tells us how likely the universe is to exist in a certain state.

To draw this map, scientists don't just look at one single history of the universe. Instead, they imagine every possible history the universe could have taken, add them all up, and see which ones dominate. This is the "path integral."

This paper, written by Shubhashis Mallik and Gaurav Narain, dives deep into the messy, complicated details of this calculation, specifically looking at what happens when the universe gets very, very big (the "Infrared" or IR limit).

Here is the story of their research, broken down with everyday analogies:

1. The Setup: The "No-Boundary" Hotel

The researchers are studying a specific type of universe proposed by Stephen Hawking and James Hartle, called the No-Boundary Proposal.

• The Analogy: Imagine the universe is a hotel. The "No-Boundary" condition means the hotel has no front door or lobby; it just starts smoothly from a single point (the "South Pole") and expands outward.
• The Goal: They want to calculate the probability of the universe expanding to a certain size or shape. To do this, they have to sum up all the possible "rooms" (geometries) the universe could occupy.

2. The Problem: The Universe is Wobbly

In their calculations, they realized they couldn't just look at a smooth, perfect universe. They had to account for fluctuations—tiny ripples and wobbles in the fabric of space-time (gravitons).

• The Analogy: Imagine trying to balance a spinning top. If the top is perfectly smooth, it's easy to predict. But if the top has tiny bumps and wobbles (fluctuations), predicting its path becomes a nightmare.
• The Twist: The researchers found that the way you describe these "bumps" matters. They tested two ways of describing them:
  1. Linear Split: Like adding small blocks to a tower.
  2. Exponential Parametrization: Like stretching a rubber sheet.
  • The Discovery: They found that the "rubber sheet" method (exponential) made the math much simpler and cleaner, turning complex, tangled rules into simple, straight lines.

3. The Journey: From Smooth to Chaotic

The universe in their model starts in a Euclidean phase (think of this as a smooth, round, 4D ball where time acts like a spatial direction) and transitions into a Lorentzian phase (our familiar expanding universe with real time).

• The Analogy: Think of a balloon being inflated. At first, it's just a smooth, round shape (Euclidean). As you blow harder, it stretches out and becomes a long, expanding tube (Lorentzian).
• The Complex Saddles: To calculate the probability, they had to look at "saddles." Imagine a horse saddle: it curves up in one direction and down in another. In math, the "dominant" paths the universe takes are these saddle points. The researchers found that the most important paths are complex (involving imaginary numbers), meaning the universe takes a "shortcut" through a mathematical dimension we can't easily visualize to get from the smooth start to the expanding end.

4. The Big Surprise: The "Secular Growth" (The Infinite Echo)

This is the most critical finding of the paper. When they calculated the probability of the universe getting very large, they found something alarming.

• The Analogy: Imagine you are in a giant cathedral and you clap your hands. The sound (the probability) should fade away. But in this universe, the sound doesn't fade; it grows louder and louder the longer you wait.
• The Science: As the universe expands, the "noise" from the quantum fluctuations (the wobbles) doesn't die out. Instead, it accumulates. This is called secular growth. It means that for a very large universe, the quantum corrections become so huge that they might overwhelm the classical picture of the universe.
• The Result: The wave function (the probability map) blows up. It suggests that the universe might be unstable or that our current understanding of gravity breaks down at these huge scales.

5. The Comparison: The "Pure" Universe vs. The "No-Boundary" Universe

To check if this "blowing up" was a weird artifact of their specific "No-Boundary" hotel, they compared it to a Pure de Sitter universe (a universe that has been expanding forever without a special "start" point).

• The Analogy: They compared the "No-Boundary" hotel to a "Pure" open field.
• The Finding: Surprisingly, both universes showed the same "blowing up" behavior. The "noise" grew in the same way. This tells the researchers that this isn't just a quirk of the "No-Boundary" idea; it's a fundamental feature of how gravity and quantum mechanics interact in an expanding universe.

6. The Fix: The "Magic Tweaker" (iϵ)

When they tried to do the math for the "Pure" universe, they hit a wall. The equations had "pinch singularities"—points where the math broke down completely, like a gear grinding to a halt.

• The Analogy: Imagine trying to drive a car over a road with a giant pothole. You can't cross it.
• The Solution: They used a mathematical trick called the $i\epsilon$ prescription. Think of this as slightly tilting the road or adding a tiny bit of "imaginary" friction. This tiny tweak allowed the car to drive around the pothole instead of crashing into it.
• The Result: With this tweak, they could calculate the probability, and they confirmed that the "growing noise" (IR divergence) was real and consistent with the No-Boundary model.

7. The Safety Check: KSW Allowability

Finally, they had to make sure these complex, imaginary paths weren't "illegal" or unphysical. There is a rule called the KSW Criterion (named after Kontsevich, Segal, and Witten) that acts like a bouncer at a club. It checks if a path is "allowable" or if it leads to nonsense.

• The Finding: They checked their complex paths and found that, thanks to their slight "tilt" (the $i\epsilon$), the paths passed the bouncer's check. They are allowed to exist in the calculation.

Summary: What Does It All Mean?

This paper is a deep dive into the "noise" of the universe.

1. Method: They used a smarter way of describing space-time wobbles (exponential parametrization) to make the math work.
2. Discovery: They found that as the universe gets huge, the quantum "noise" doesn't stop; it grows uncontrollably.
3. Implication: This growth happens whether the universe started with a "No-Boundary" or just expanded forever. This suggests that our current theories might need a major upgrade to handle the universe when it gets truly massive.
4. Conclusion: The universe is stable enough to exist, but its quantum fluctuations are louder and more persistent than we thought, echoing endlessly as it expands.

In short: The universe is expanding, but the quantum "static" on the radio is getting louder and louder, and we need to figure out how to tune it out before the signal gets lost.

Technical Summary

1. Problem Statement

The paper investigates the infrared (IR) properties of the gravitational path integral in 4D Einstein-Hilbert gravity with a positive cosmological constant ($\Lambda$). Specifically, it aims to compute the Hartle-Hawking (H-H) wave function of the Universe to one-loop order for complex geometries with topology $\mathbb{R} \times S^3$.

Key challenges addressed include:

• Boundary Conditions: Determining consistent boundary conditions for both the background scale factor and metric fluctuations ($h_{ij}$) that respect general covariance.
• Parametrization Dependence: Analyzing how the choice of parametrization for metric fluctuations (linear split vs. exponential) affects allowed boundary conditions and the resulting action.
• IR Divergences: Understanding the behavior of the wave function as the Universe expands (the deep IR limit), specifically looking for secular growth (divergences) arising from graviton fluctuations.
• Comparison: Comparing the IR behavior of the "No-Boundary" (complex saddle) proposal against pure Lorentzian de Sitter (dS) geometries.

2. Methodology

The authors employ a rigorous semi-classical approach using the Lorentzian gravitational path integral.

• Metric Ansatz: The spacetime metric is decomposed using the ADM formalism with a specific ansatz for fluctuations:
  $$ds^2 = -N^2(t)dt^2 + a^2(t) \left[ \rho_{ij} + h_{ij} + \frac{\epsilon}{2} h_{ik}h^k_j + \dots \right] dx^i dx^j$$
  Here, $\epsilon$ is a parametrization parameter: $\epsilon=0$ corresponds to the linear split, and $\epsilon=1$ corresponds to the exponential parametrization.
• Boundary Conditions:
  • Initial Boundary (No-Boundary): Imposed as a Robin condition (linear combination of scale factor and conjugate momentum) to ensure stable Gaussian fluctuations.
  • Final Boundary: Two scenarios are considered:
    1. Fixed Size (Dirichlet): Fixed final scale factor $q_f$.
    2. Fixed Extrinsic Curvature (Conformal): Fixed Hubble rate $H_1$ (non-linear condition).
• Gauge Fixing & Ghosts: The transverse-traceless (TT) gauge is used. The Faddeev-Popov procedure is applied to handle diffeomorphism redundancy, introducing ghost fields.
• Path Integral Computation:
  • Background: Integrated exactly for linear boundary conditions.
  • Fluctuations: Computed up to one-loop using the Gel'fand-Yaglom method to evaluate functional determinants of Sturm-Liouville operators.
  • Lapse Integration: The integral over the lapse function $N_c$ is evaluated using the Picard-Lefschetz (PL) theory and WKB approximation to handle the highly oscillatory nature of the Lorentzian integral.
• Regularization & Renormalization:
  • UV Divergences: Extracted and canceled using generalized Hurwitz-Zeta function regularization and appropriate counterterms.
  • IR Analysis: Asymptotic expansions are performed in the limit of large spatial volume ($q_f \to \infty$ or $H_1 \to H$).
• Stability & Allowability: The KSW (Kontsevich-Segal-Witten) criterion is used to check if the complex saddle geometries are "allowable" (physically consistent) by ensuring the real part of the Euclidean action for $p$-forms is positive.

3. Key Contributions

1. Parametrization and Boundary Conditions:

   • The authors demonstrate that general covariance tightly constrains the allowed boundary conditions for fluctuations based on the background choice.
   • Crucially, they find that for the exponential parametrization ($\epsilon=1$), the allowed boundary conditions for fluctuations become linear (specifically, fixing $h_{ij}$ is allowed even with fixed extrinsic curvature). In contrast, the linear split ($\epsilon=0$) leads to complicated non-linear boundary conditions. This suggests the exponential parametrization is superior for cosmological path integrals.

2. One-Loop Renormalized Action:

   • They explicitly compute the one-loop effective action for the lapse, including contributions from background fluctuations, graviton modes, and ghosts.
   • They derive specific counterterms required to cancel UV divergences arising from the vanishing initial size (No-Boundary) and the contact limit in de Sitter space.

3. KSW Allowability of Fixed-K Saddles:

   • They prove that the complex No-Boundary saddles with fixed extrinsic curvature satisfy the KSW allowability criterion, confirming their physical validity in the path integral.

4. Handling IR Divergences in Pure de Sitter:

   • For pure Lorentzian de Sitter (real saddles), the authors encounter "pinch" singularities in the one-loop determinant.
   • They introduce a novel $i\epsilon$-prescription achieved by slightly complexifying the cosmological constant ($\Lambda \to \Lambda e^{i\bar{\epsilon}}$). This shifts the poles off the real axis, regularizing the functional determinant and allowing the PL method to yield a convergent result.

4. Key Results

• Secular Growth (IR Divergence):

  • In the deep IR limit (large Universe), the one-loop corrected wave function exhibits secular growth. The amplitude grows exponentially with the spatial volume of the hypersurface:
    $$\Psi \sim \exp\left( \frac{\pi}{12\sqrt{2}} \frac{1}{(1-H_1)^{3/2}} \right)$$
  • This growth is driven by the graviton fluctuations ($h_{ij}$) and dominates over the semiclassical background contribution.
  • This behavior is universal: it appears in both the No-Boundary (complex saddle) and Pure de Sitter (real saddle) scenarios, regardless of whether the final boundary condition is fixed size or fixed curvature.

• Phase Structure:

  • For the No-Boundary case, the wave function is real due to the anti-linear symmetry of the two contributing complex saddles ($N_+ = -N_-^*$).
  • For the Pure de Sitter case with the $i\epsilon$ deformation, the wave function becomes complex because the saddles are no longer complex conjugates.

• Parametrization Independence:

  • Despite the dependence of intermediate steps on $\epsilon$, the final renormalized on-shell action and the resulting wave function are independent of the parametrization ($\epsilon$), consistent with general theorems in effective field theory.

5. Significance

• Validation of No-Boundary Proposal: The study confirms that complex No-Boundary saddles are physically allowable (KSW) and stable against metric fluctuations, even with non-linear boundary conditions like fixed extrinsic curvature.
• Universality of IR Growth: The finding that IR secular growth occurs in both complex (No-Boundary) and real (Pure dS) geometries suggests that this divergence is an intrinsic feature of quantum gravity in de Sitter space, rather than an artifact of the complex contour choice.
• Methodological Advancement: The paper provides a robust framework for handling one-loop determinants in Lorentzian quantum cosmology, particularly the use of exponential parametrization to simplify boundary conditions and the $i\epsilon$-deformation of $\Lambda$ to resolve pinch singularities in real geometries.
• Implications for Cosmology: The secular growth implies that perturbative expansions in de Sitter space may break down at late times, suggesting that non-perturbative effects or resummation techniques might be necessary to describe the late-time universe accurately.

In summary, this work provides a comprehensive one-loop analysis of the Hartle-Hawking wave function, establishing the robustness of the No-Boundary proposal under various boundary conditions and parametrizations, while highlighting a universal infrared instability in de Sitter quantum gravity.