Relic gravitational waves from primordial gravitational collapses

This paper presents a hybrid numerical analysis demonstrating that the collision of sound shells generated by large primordial density perturbations produces a stochastic gravitational wave background with detectable signatures for future observatories, offering new constraints on primordial black holes even after they have evaporated.

The Gist

Imagine the early Universe not as a smooth, empty void, but as a giant, bubbling pot of cosmic soup. In this soup, tiny ripples of density appeared—some spots were slightly denser (heavier), and some were slightly less dense (lighter) than the average.

This paper, written by a team of physicists, explores what happens when these ripples get really big. They ask: "If a ripple is massive enough, does it just collapse into a black hole, or does it do something else?"

Here is the story of their discovery, broken down into simple concepts and analogies.

1. The Cosmic "Boing!" (The Sound Shell)

Usually, when a massive object collapses, we think of it imploding into a black hole. But the authors found that even if a ripple doesn't collapse into a black hole, it still does something dramatic.

Imagine you are in a crowded room and someone suddenly pushes a giant wave of people outward. The people in the center get pushed away, leaving an empty space behind them, while a thick ring of people rushes outward.

In the early Universe, when a large density ripple forms:

• The Center: The fluid (the "soup") gets pushed away, creating a low-density bubble.
• The Edge: A thick, expanding shell of high-density fluid rushes outward.

The authors call this an "Outgoing Sound Shell." It's like a giant, cosmic shockwave or a "Boing!" sound traveling through the fabric of space.

2. The Great Cosmic Collision

The Universe was filled with billions of these ripples. As time passed, these expanding "sound shells" grew larger and larger until they crashed into each other.

Imagine a giant pool party where everyone is blowing up a balloon. Eventually, the balloons bump into each other. When these cosmic sound shells collide, they don't just bounce off; they create a massive, violent disturbance.

This collision is the key. The authors calculated that these collisions generate Gravitational Waves.

• What are Gravitational Waves? Think of them as ripples in a pond, but instead of water, it's the fabric of space and time itself.
• The Analogy: If the Universe were a trampoline, and you dropped a bowling ball on it, it would wiggle. If you dropped millions of bowling balls and they all hit the trampoline at once, the whole thing would shake violently. That shaking is the gravitational wave.

3. The "Hybrid" Detective Work

How did they figure this out? They didn't just use math on a piece of paper. They used a "hybrid" approach:

1. Supercomputer Simulations: They ran complex video-game-style simulations of the early Universe to see exactly how these sound shells move and crash.
2. Mathematical Models: They took the data from those simulations and plugged it into a theoretical model (the "Sound Shell Model") to predict what kind of gravitational waves would be produced.

They found that the "sound" of these collisions creates a specific pattern of gravitational waves that is different from other known sources (like colliding black holes today).

4. Why Should We Care? (The Treasure Hunt)

The paper suggests that these ancient waves are still traveling through the Universe today. They are part of the Stochastic Gravitational Wave Background (SGWB). This is like a constant, low-level hum of the Universe, made up of the echoes of billions of these ancient collisions.

The Detective Tools:
The authors predict that future "ears" (detectors) might be able to hear this hum:

• Pulsar Timing Arrays (PTAs): These use spinning stars (pulsars) as cosmic clocks to detect very low-frequency waves.
• Space Detectors (like LISA, Taiji, TianQin): These are satellites designed to listen to higher frequencies.
• Ground Detectors (like LIGO, Einstein Telescope): These listen to the highest frequencies.

The "Smoking Gun":
If we detect this specific type of gravitational wave background, it tells us two huge things:

1. Primordial Black Holes (PBHs): It confirms that tiny black holes formed in the very first second of the Universe.
2. The "Ghost" Black Holes: Some of these black holes might have been so small they evaporated (disappeared) billions of years ago due to Hawking radiation. We can't see them anymore, but their "sound shells" left a permanent scar on the Universe in the form of gravitational waves. Detecting the waves is the only way to prove these "ghost" black holes ever existed.

Summary

• The Event: Big ripples in the early Universe created expanding shells of sound.
• The Action: These shells crashed into each other, shaking the fabric of space.
• The Result: A unique "hum" of gravitational waves that is still traveling today.
• The Goal: Future detectors might hear this hum, giving us a new way to find evidence of tiny, ancient black holes that have long since vanished.

In short, this paper suggests that the Universe is still ringing like a bell from a massive collision that happened 13.8 billion years ago, and we are finally building the ears to hear it.

Technical Summary

1. Problem Statement

The paper addresses the generation of the Stochastic Gravitational Wave Background (SGWB) resulting from the gravitational collapse of large primordial density perturbations in the early Universe.

• Context: While it is well-established that sufficiently large perturbations lead to Primordial Black Hole (PBH) formation, perturbations that are large but sub-critical (or near-critical) do not form black holes. Instead, they undergo a "bounce," generating an outgoing shell of density contrast known as a sound shell.
• Gap in Knowledge: Previous studies of Scalar-Induced Gravitational Waves (SIGWs) typically rely on perturbation theory. However, the dynamics of sound shell formation and collision involve inherently non-perturbative effects (large amplitude fluid motions) that standard perturbative approaches fail to capture accurately.
• Goal: To provide a rigorous, hybrid numerical analysis of the SGWB induced by the collision of these sound shells, regardless of whether a PBH is formed, and to determine the resulting GW spectrum's peak frequency and amplitude.

2. Methodology

The authors employ a hybrid approach combining numerical relativity simulations with an analytical sound shell model.

A. Numerical Simulations

• Framework: The authors solve the Misner-Sharp equations for a relativistic perfect fluid in a spherically symmetric spacetime during the radiation-dominated (RD) era.
• Initial Conditions: They assume a Gaussian profile for the initial curvature perturbation $\zeta(r) = \mu e^{-(r/r_m)^2}$.
• Scenarios: Simulations cover three regimes based on the perturbation amplitude $\mu$:
  1. Sub-critical ($\mu = 0.4$): No PBH forms; an outgoing sound wave with both overdense and underdense shells forms.
  2. Near-critical ($\mu = 0.8$): PBH formation occurs, but a complex structure with both underdense and overdense shells is generated due to large pressure gradients.
  3. Super-critical ($\mu = 0.9$): PBH formation occurs; only an underdense shell propagates outward.
• Data Extraction: The simulations track the evolution of energy density, density contrast ($\delta$), and comoving radial velocity ($u$) up to the point of sound shell collision.

B. Analytical Modeling (Sound Shell Model)

• Input: The authors extract the fluid velocity profiles ($u(r)$) and density contrasts ($\delta(r)$) from the numerical simulations just before the sound shells collide.
• Velocity Power Spectrum: They derive the velocity power spectrum $\langle v_i v_j^* \rangle$ using the extracted profiles. The spectrum is defined by a function $A(q)$, which combines the Fourier transforms of the velocity and density profiles.
• GW Spectrum Calculation: Using the sound shell model (adapted from first-order phase transition literature), they calculate the induced GW power spectrum $P_{GW}$. The spectrum depends on the velocity power spectrum, the Hubble horizon at formation, and the mean separation of the sound shells ($R^*_c$).
• Key Formula: The present-day GW energy density is given by:
  $$h^2\Omega_{GW}(f) \propto \left(\frac{r_H}{R^*_c}\right)^7 \tilde{P}_{GW}(k r_H)$$
  where the strong dependence on the shell separation ($R^*_c$) highlights the importance of the abundance of perturbations.

3. Key Contributions

1. First Non-Perturbative Treatment: This work is the first to capture the inherently non-perturbative dynamics of sound shell sources in the context of primordial collapse, moving beyond the limitations of second-order perturbation theory used in standard SIGW calculations.
2. Hybrid Numerical-Analytical Framework: The paper successfully bridges high-fidelity hydrodynamic simulations with the analytical sound shell model, validating the analytical predictions against numerical data.
3. Unified Source Description: It demonstrates that sound shells are generated in all cases of large density perturbations (sub-critical, near-critical, and super-critical), meaning GWs are produced even if no PBHs survive to the present day.
4. Differentiation of PBH Signatures: The study identifies distinctive features in the GW spectra that could potentially distinguish between PBHs formed from super-critical collapses versus those formed from near-critical collapses.

4. Key Results

• GW Spectrum Characteristics:
  • The GW spectrum exhibits a peak frequency determined by the thickness of the underdense shell at the time of collision.
  • The amplitude is highly sensitive to the abundance of sound shells (mean separation $R^*_c$). A larger separation (lower abundance) drastically reduces the GW amplitude due to the $(r_H/R^*_c)^7$ scaling factor.
  • Peak Frequency ($f_0$):
    • For asteroid-mass PBHs ($M_{PBH} \sim 10^{20}$ g), $f_0 \sim 10^{-2}$ Hz, falling within the sensitivity range of space-based detectors like LISA, Taiji, TianQin, DECIGO, and BBO.
    • For extremely light PBHs ($M_{PBH} < 10^{10}$ g), $f_0 \sim 10^3$ Hz, which is currently out of reach for ground-based detectors (LIGO/Virgo) but could be probed by future high-frequency detectors.
    • For heavier PBHs ($10^{10} - 10^{13}$ g), $f_0 \sim 10 - 100$ Hz, within the LIGO-Virgo-KAGRA band.
• Comparison with SIGWs:
  • The Supplemental Material shows that for large positive curvature amplitudes ($\mu > 0.4$), the acoustic GWs generated by sound shell collisions can be up to 10 times larger than the standard scalar-induced GWs (SIGWs) calculated via perturbation theory.
• Observational Constraints:
  • Future null detections of high-frequency GW backgrounds could place new, stringent constraints on the abundance ($\beta$) of PBHs that have already evaporated via Hawking radiation.
  • Specifically, if no signal is detected, the PBH abundance $\beta$ could be constrained to be smaller than $\mathcal{O}(10^{-5})$.

5. Significance

• New Probe for Evaporated PBHs: This mechanism offers a unique observational window into PBHs that have completely evaporated, which are otherwise invisible to traditional electromagnetic or dynamical constraints.
• Multi-Messenger Cosmology: The predicted signals span a wide frequency range, making them detectable by the next generation of GW observatories (PTAs, LISA, ET, CE, and high-frequency MHz-GHz detectors).
• Theoretical Advancement: By validating the sound shell model against non-perturbative simulations, the paper provides a more robust theoretical foundation for interpreting future SGWB data, potentially revealing the physics of the early Universe's density fluctuations and the nature of dark matter candidates.

In summary, the paper establishes that the collision of sound shells from primordial collapses is a potent source of relic gravitational waves, offering a powerful new tool to constrain the abundance and mass distribution of primordial black holes, including those that no longer exist.