Cosmological angular momentum from quantum rotation

This paper proposes that cosmic angular momentum, particularly the high spins of primordial black holes, originates from quantum fluctuations of a rotating spectator scalar field during inflation, which generate spatial momentum flows that convert into net rotation during gravitational collapse, offering a testable alternative to tidal torque theory.

The Gist

The Big Mystery: Why Does Everything Spin?

Imagine looking up at the night sky. You see galaxies swirling like giant pinwheels, and black holes spinning like cosmic tops. The universe is full of rotation. But here's the puzzle: Where did all that spin come from?

For decades, scientists have used a theory called "Tidal Torque Theory" to explain this. Think of it like a child on a swing. The swing (a galaxy) doesn't start moving on its own; it gets pushed by the wind (gravity from nearby stars and dark matter) that pushes it off-center. This theory works well for big things like galaxies, predicting they spin slowly.

But there's a problem. This theory struggles to explain Primordial Black Holes (PBHs)—tiny black holes that might have formed right after the Big Bang. If they formed from a perfectly round collapse of gas, they shouldn't spin at all. Yet, if they do spin fast, it changes how they behave, how they merge, and how we detect them with gravitational waves.

The New Idea: The "Quantum Ice Skater"

The author, Bo-Qiang Lu, proposes a brand new mechanism. Instead of waiting for gravity to push things into a spin, he suggests that the spin was built-in from the very beginning, born from the quantum fluctuations of the early universe.

Here is the step-by-step analogy of how this works:

1. The Invisible Spinning Top (The Field)

Imagine the early universe was filled with an invisible, invisible fluid called a "complex scalar field." Think of this field like a giant, invisible ice skater spinning on a frozen lake.

• This skater has a "charge" (like a score on a scoreboard) that represents their internal spin.
• Normally, if the skater spins perfectly in place without moving across the ice, they have internal energy, but they aren't moving through space.

2. The Quantum Ripples (Inflation)

During "Inflation" (a period where the universe expanded faster than light), this ice skater wasn't perfectly smooth. Quantum mechanics made them wobble.

• Imagine tiny ripples appearing on the ice surface around the skater.
• These ripples represent density fluctuations. Some parts of the ice are slightly higher, some slightly lower.

3. The "Kick" (Breaking Symmetry)

The paper suggests that the laws of physics (specifically, a broken symmetry) gave this skater a little "kick" or a nudge. This started them rotating in a specific direction, storing up a massive amount of internal angular momentum (like a flywheel spinning at high speed).

4. The Crash Course (Horizon Re-entry)

As the universe expanded and then slowed down, these quantum ripples grew larger and eventually "fell back" into our observable universe (re-entered the horizon).

• Here is the magic trick: The ripples (the uneven ice) interacted with the spinning skater (the background field).
• Because the ice was uneven, the skater couldn't just spin in place anymore. The unevenness forced the skater to slide.
• The Conversion: The internal spin of the skater was converted into physical movement across the ice. The "spin" became a "flow."

5. The Collapse (The Black Hole is Born)

Now, imagine a patch of this ice where the skater is sliding and the ice is collapsing inward to form a hole.

• If the ice collapses perfectly into a sphere, the sliding cancels out, and no spin remains.
• BUT, if the ice collapses into a squashed oval (an ellipsoid)—which is very common in nature—the sliding motion gets trapped.
• The "flow" of the skater gets twisted into a net spin.
• When this patch collapses to form a Primordial Black Hole, the black hole is born already spinning, carrying the momentum from that ancient quantum skater.

Why Does This Matter?

1. Super Fast Spins
The old theory (Tidal Torque) predicts these black holes should spin very slowly (like a lazy turnstile). This new mechanism predicts they could be spinning extremely fast (like a bullet train).

• The Result: The "spin parameter" (how fast it spins) could be anywhere from 0.1 to 1.0 (where 1 is the speed limit of a black hole). This is huge compared to the old predictions.

2. A Testable Clue
If we can detect these black holes using gravitational wave detectors (like LIGO, Virgo, or the future LISA), we can measure their spin.

• If we find black holes spinning near the speed limit, it's a "smoking gun" that proves this quantum mechanism is real.
• It would mean we are directly observing the quantum fluctuations of the Big Bang converted into the spin of a black hole.

The Bottom Line

This paper suggests that the spin of the universe's smallest black holes isn't just a result of them bumping into each other later on. Instead, it's a fossil record of the universe's birth.

It's like finding a seashell on a mountain and realizing it proves the mountain was once underwater. Similarly, finding a super-spinning black hole would prove that the universe's rotation started as a tiny quantum wobble that got stretched, twisted, and trapped during the formation of the first black holes.

In short: The universe didn't just get pushed into a spin; it was born spinning, and this paper explains exactly how the quantum mechanics of the Big Bang wrote that spin into the fabric of space itself.

Technical Summary

1. Problem Statement

The origin of cosmic angular momentum (spin) in structures like galaxies and black holes (BHs) is a fundamental unsolved question in cosmology.

• Current Paradigm: The standard explanation is Tidal Torque Theory (TTT). TTT posits that proto-halos acquire spin due to the misalignment between their inertia tensor and the external tidal gravitational field generated by neighboring large-scale structures.
• Limitations of TTT:
  • It describes the conversion of existing asymmetries into spin, not the fundamental creation of angular momentum (which ultimately traces back to quantum fluctuations).
  • It predicts low spin parameters ($\chi \sim 0.01 - 0.1$) for dark matter halos and very low spins ($\chi \sim 10^{-3} - 10^{-2}$) for Primordial Black Holes (PBHs) formed in a radiation-dominated universe.
  • It struggles to explain the extremely high spins inferred for some supermassive black holes without invoking complex secondary processes (e.g., major mergers, gas accretion).
• The Gap: There is no established mechanism linking the quantum origin of structure (inflationary fluctuations) directly to the intrinsic spin of collapsed objects like PBHs.

2. Methodology and Mechanism

The author proposes a novel mechanism where spatial angular momentum is generated directly from the internal rotation of a spectator complex scalar field during inflation.

Theoretical Framework

• Field Setup: A complex scalar field $P$ with a global $U(1)$ symmetry (e.g., Peccei-Quinn symmetry) is introduced.
  • Lagrangian: $\mathcal{L} = \partial_\mu P \partial^\mu P^* - V(|P|^2)$.
  • Parameterization: $P = \frac{1}{\sqrt{2}}S e^{i\theta}$, where $S$ is the radial mode and $\theta$ is the angular (axion-like) mode.
• Internal Angular Momentum: The field possesses a conserved $U(1)$ charge density ($n_c = S^2 \dot{\theta}$), representing internal angular momentum. This rotation is triggered by higher-dimensional operators that explicitly break the $U(1)$ symmetry (providing an initial "angular kick").
• Inflationary Dynamics:
  • During inflation, quantum fluctuations stretch the field to superhorizon scales, creating spatial perturbations in both the radial ($\delta S$) and angular ($\delta \theta$) components.
  • The radial mode $S$ is assumed to be heavy or evolve slowly, sitting in a steep potential, while the angular mode rotates.
• Conversion Mechanism (Horizon Re-entry):
  • When a perturbation mode re-enters the horizon, spatial gradients ($\partial_i \delta \theta$) couple to the large background charge density ($S_0^2 \dot{\theta}_0$).
  • This coupling generates a non-zero momentum density ($T_{0i}$) in the energy-momentum tensor:
    $$T_{0i} \approx S_0^2 \dot{\theta}_0 \partial_i \delta \theta$$
  • This converts the internal field rotation into a bulk spatial momentum flow.

Gravitational Collapse Analysis

The paper analyzes how this momentum flow translates into the spin of a collapsing object (PBH):

• Spherical Collapse: The authors prove mathematically that for a perfectly spherical collapse, the total angular momentum vanishes ($J_i = 0$) regardless of the phase perturbation distribution. The symmetry of the sphere cancels out the momentum flow.
• Non-Spherical (Ellipsoidal) Collapse:
  • If the collapse is non-spherical (e.g., ellipsoidal), the geometric anisotropy allows the momentum flow to generate net angular momentum.
  • The angular momentum $J$ depends on the coupling between the quadrupole moment of the phase perturbation ($Q_{jk}$) and the ellipticity of the collapsing region.
  • Formula derived: $J_i \propto S_0^2 \dot{\theta}_0 \cdot Q_{jk} \cdot (\text{Moment of Inertia Difference})$.

3. Key Contributions

1. Fundamental Origin of Spin: Establishes a direct link between the quantum origin of structure (inflationary fluctuations) and the macroscopic spin of collapsed objects, bypassing the limitations of TTT.
2. Mechanism of Conversion: Demonstrates how internal $U(1)$ charge density (field rotation) is converted into spatial angular momentum via inflationary spatial gradients upon horizon re-entry.
3. Geometric Necessity: Proves that non-spherical collapse is a strict prerequisite for this mechanism to generate net spin; purely spherical collapses yield zero spin.
4. Spin-Abundance Correlation: Derives a quantitative relationship showing that the dimensionless spin parameter ($\chi$) of PBHs is directly determined by the amplitude of the primordial power spectrum on small scales.

4. Results

• Spin Estimation:
  • For a scale-invariant power spectrum (standard inflation), the predicted PBH spin is negligible ($\chi \sim 10^{-5}$).
  • However, if the power spectrum is enhanced on small scales (required to produce a detectable abundance of PBHs consistent with LIGO-Virgo-KAGRA observations, i.e., $\mathcal{P}_\zeta \sim 10^{-2} - 10^{-1}$), the spin is significantly boosted.
• Predicted Spin Values:
  • Under conditions of enhanced small-scale power, the mechanism predicts PBH spins in the range $\chi \sim 0.1 - 1$.
  • This is orders of magnitude higher than TTT predictions ($\chi \sim 10^{-3}$) and approaches the maximum theoretical limit for astrophysical black holes ($\chi_{lim} \approx 0.998$).
• Distribution: The mechanism predicts a spin distribution linked to the small-scale power spectrum, potentially exhibiting bimodal or broad shapes with high-spin tails.

5. Significance and Implications

• Testable Prediction: The mechanism offers a distinct, testable signature for gravitational wave (GW) astronomy. If PBHs are detected with high spins ($\chi > 0.1$), it would strongly support this quantum-rotation origin over standard tidal torque models.
• Violation of Thorne Limit: If PBHs are found with $\chi > 0.998$, it would violate the Thorne limit for astrophysical BHs (which assumes gas accretion), providing "smoking gun" evidence for a primordial, quantum-origin spin.
• Phenomenological Consequences: High-spin PBHs would exhibit:
  • Enhanced Hawking radiation (affecting Big Bang Nucleosynthesis and CMB).
  • Stronger superradiance effects (potentially detectable via GWs or dark matter signatures).
  • Modified merger rates and GW waveforms detectable by current (LVK) and future (LISA, Einstein Telescope) observatories.
• Broader Applicability: If the rotation field constitutes a significant fraction of Dark Matter, this mechanism could also explain the high spins observed in some supermassive black holes and dark matter halos without relying on complex merger histories.

In summary, the paper proposes a paradigm shift where the spin of cosmic structures is not merely a result of tidal interactions but a direct imprint of quantum field rotation during inflation, modulated by the geometry of gravitational collapse and the amplitude of primordial density fluctuations.