Gaussian Planck Relics are Ruled-Out as Dark Matter by LIGO

This paper demonstrates that LIGO's constraints on the gravitational wave background rule out the formation of Planck star relics as dark matter from Gaussian initial conditions, leaving non-Gaussian primordial fluctuations as the sole viable formation channel.

The Gist

The Big Idea: The Universe's "Unbreakable" Leftovers

Imagine the universe is a giant construction site. Sometimes, matter gets crushed so hard by gravity that it collapses into a black hole. According to old-school physics (Einstein), this matter gets crushed into an infinitely small, infinitely dense point called a singularity—a place where the laws of physics break down.

But new theories in Quantum Gravity (specifically Loop Quantum Gravity) suggest that nature has a safety net. When matter gets crushed to the absolute limit (the "Planck scale"), it doesn't disappear into a singularity. Instead, it hits a "quantum trampoline" and bounces back.

Because this happens inside the black hole's event horizon, we can't see the bounce. To the outside world, it looks like a black hole that has evaporated away. But inside, a tiny, stable, indestructible speck of matter remains. The authors call these "Planck Star Remnants" (PSRs).

The Big Question: Could these tiny, invisible specks be the Dark Matter that holds our galaxies together?

The Investigation: A Cosmic "Smoking Gun"

The authors (Oem Trivedi and Abraham Loeb) decided to test this idea. They asked: If the universe is full of these Planck remnants, what evidence would we see?

They realized that for the universe to have enough of these remnants to make up all the Dark Matter, a huge number of tiny black holes must have formed in the very early universe.

Here is the problem: Forming black holes is loud.

Think of the early universe like a calm pond. If you drop a pebble (a small fluctuation in density), you get a tiny ripple. But to create a black hole, you need a massive boulder to crash into the water. That crash creates a huge splash and sends out powerful shockwaves. In physics, these shockwaves are Gravitational Waves (ripples in space-time).

The Conflict: The "Silent" Detector vs. The "Noisy" Theory

The authors ran the numbers to see how many black holes would need to form to create enough Dark Matter.

1. The Gaussian Scenario (The "Normal" Way):
   Imagine the density fluctuations in the early universe were like a standard bell curve (a "Gaussian" distribution). Most fluctuations are small, and huge ones are incredibly rare.

   • To get enough black holes to make Dark Matter, the universe would have needed massive fluctuations.
   • The Result: These massive fluctuations would have created a deafening roar of gravitational waves.
   • The Evidence: We have a very sensitive "ear" for these waves: LIGO (the observatory that detected black hole collisions). LIGO has listened to the universe and said, "We hear a very quiet background hum, but we do not hear the deafening roar that this theory predicts."
   • The Verdict: The "Normal" (Gaussian) way of making these Planck remnants is ruled out. The noise would have been too loud, and LIGO would have heard it.

2. The Non-Gaussian Scenario (The "Weird" Way):
   Is there a way to make the black holes without making the noise?

   • Imagine the fluctuations weren't a smooth bell curve, but had "heavy tails" (like a distribution where extreme events happen more often than you'd expect).
   • In this scenario, you can get the same number of black holes (the "boulders") without needing the average wave height to be as high.
   • The Result: This creates the necessary Dark Matter but keeps the gravitational wave background quiet enough to pass LIGO's test.

The Conclusion: A New Clue for the Universe

The paper concludes with a clear message:

• If Dark Matter is made of these "Planck Star Remnants," then the early universe must have been weird.
• The fluctuations that created them could not have been "normal" (Gaussian). They must have been Non-Gaussian (having those "heavy tails" or extreme outliers).

The Metaphor:
Imagine you walk into a room and find a pile of broken glass (Dark Matter).

• Theory A (Gaussian): You assume a gentle breeze blew a vase off the shelf. But a gentle breeze wouldn't break that many vases. To break them all, you'd need a hurricane. But if there was a hurricane, the windows would be shattered, and the neighbors (LIGO) would have called the police. Since the windows are fine, Theory A is wrong.
• Theory B (Non-Gaussian): You assume someone threw a few specific, heavy rocks at the vase. This breaks the glass (creates Dark Matter) without causing a hurricane (Gravitational Waves). This fits the evidence.

Why This Matters

This paper doesn't say Planck Star Remnants can't be Dark Matter. Instead, it acts as a filter. It tells us that if they are the answer, the universe's history is stranger than we thought. It suggests that the early universe wasn't a smooth, predictable place, but one where rare, extreme events happened frequently enough to build our Dark Matter without screaming too loudly for our telescopes to hear.

In short: The "quiet" universe we see today rules out the "loud" way of making Planck remnants. If they exist, they must have been born from a chaotic, non-standard beginning.

Technical Summary

1. Problem Statement

The paper addresses the viability of Planck Star Remnants (PSR) as a candidate for Cold Dark Matter (CDM).

• Theoretical Context: In Loop Quantum Gravity (LQG), specifically Loop Quantum Cosmology (LQC), gravitational singularities are replaced by "quantum bounces." When a black hole evaporates via Hawking radiation down to the Planck mass ($M_{Pl} \sim 10^{-5}$ g), quantum geometry effects prevent total evaporation, leaving behind a stable, non-singular relic of Planck mass.
• The Hypothesis: Primordial Black Holes (PBHs) formed in the early universe could evaporate and leave behind these PSRs, potentially accounting for the observed dark matter density.
• The Conflict: To explain the current dark matter density, a specific fraction of the early universe's radiation energy must collapse into PBHs. The authors investigate whether the Gaussian statistics of primordial density fluctuations can generate enough PBHs to satisfy this requirement without violating observational constraints, specifically the stochastic gravitational wave (GW) background limits set by LIGO.

2. Methodology

The authors employ a multi-step analytical approach combining cosmological perturbation theory, black hole thermodynamics, and observational constraints:

1. Relic Abundance Calculation:

   • They calculate the required collapse fraction, $\beta_{req}$, of radiation energy density into PBHs during the radiation-dominated era. This fraction is derived by equating the present-day relic number density ($n_{relic} = \rho_{DM,0}/M_{Pl}$) to the initial PBH formation rate, accounting for the expansion of the universe and the mass ratio between the horizon mass at formation and the Planck mass.
   • They identify a viable formation temperature window ($T_f \sim 10^9 - 10^{11}$ GeV) where $\beta_{req} \sim 10^{-5}$.

2. Gaussian Statistics Analysis:

   • Assuming primordial density perturbations follow a Gaussian distribution, they use the Press-Schechter formalism to relate the collapse fraction $\beta$ to the variance of the density contrast, $\sigma^2$, and the critical overdensity $\delta_c \approx 0.45$.
   • They link $\sigma^2$ to the amplitude of the primordial curvature power spectrum, $P_R(k)$.
   • They determine the required $P_R(k)$ amplitude to achieve the necessary $\beta_{req}$.

3. Gravitational Wave Induction:

   • Large curvature perturbations required to form PBHs inevitably induce a second-order stochastic gravitational wave background when these perturbations re-enter the horizon.
   • The authors calculate the peak amplitude of this induced GW spectrum ($\Omega_{GW}$) based on the required $P_R(k)$ amplitude.

4. Observational Comparison:

   • They compare the calculated $\Omega_{GW}$ against the upper limits set by LIGO O3 observations ($\Omega_{GW} \lesssim 5 \times 10^{-9}$ in the 100–1000 Hz band).

5. Non-Gaussian Alternative:

   • To resolve the tension, they explore non-Gaussian statistics (specifically lognormal and power-law distributions with heavy tails) to see if they can yield the required $\beta$ with a lower $P_R(k)$, thereby suppressing the GW signal.

3. Key Contributions

• Quantitative Constraint on Gaussian PSRs: The paper provides a rigorous derivation showing that if PSRs constitute dark matter and primordial fluctuations are Gaussian, the required curvature power spectrum amplitude ($P_R \sim 10^{-2} - 10^{-1}$) is incompatible with current GW data.
• LIGO as a Discriminator: It establishes LIGO's stochastic GW background limits as a definitive "smoking gun" test for the Gaussian formation channel of Planck mass relics.
• Non-Gaussian Necessity: The authors demonstrate that non-Gaussianity is not just a possibility but a requirement for this dark matter scenario to survive. They show that heavy-tailed distributions (lognormal or power-law) can produce the necessary collapse fraction with a much lower power spectrum amplitude ($P_R \sim 10^{-3}$), keeping the induced GW signal below LIGO's detection threshold.

4. Key Results

• The Gaussian Exclusion:

  • To achieve the required collapse fraction $\beta \sim 10^{-5}$ under Gaussian statistics, the curvature power spectrum must have a peak amplitude of $P_R(k_c) \approx 5.6 \times 10^{-2}$.
  • This amplitude induces a stochastic GW background with a peak amplitude of $\Omega_{GW} \sim 10^{-7} - 10^{-6}$ at frequencies $f \sim 100 - 1000$ Hz.
  • Conclusion: This predicted signal exceeds the LIGO O3 bound ($\sim 5 \times 10^{-9}$) by 10 to 1000 times. Therefore, the Gaussian formation pathway for Planck relics as dark matter is ruled out.

• The Non-Gaussian Solution:

  • If the primordial fluctuations follow a lognormal or power-law distribution (heavy tails), the exponential suppression of the collapse fraction seen in Gaussian cases is avoided.
  • With these distributions, a much lower power spectrum amplitude ($P_R \approx 10^{-3}$) is sufficient to generate the required $\beta \sim 10^{-5}$.
  • At this lower amplitude, the induced GW background drops to $\Omega_{GW} < 5 \times 10^{-9}$, making the scenario consistent with LIGO observations.

5. Significance

• Theoretical Implications: The paper fundamentally shifts the landscape for Planck relic dark matter. It suggests that if such relics exist, the early universe must have possessed significant primordial non-Gaussianity. This challenges standard slow-roll inflation models which typically predict nearly Gaussian fluctuations.
• Observational Strategy: It highlights the power of GW astronomy (specifically LIGO/Virgo/KAGRA) in constraining early-universe physics and dark matter candidates, moving beyond direct detection or lensing constraints.
• Future Directions: The work narrows the search space for Planck star dark matter to scenarios involving non-standard inflationary histories or mechanisms that generate heavy-tailed density perturbations. It effectively closes the "Gaussian window" for this specific dark matter candidate.

In summary, Trivedi and Loeb demonstrate that while Planck star remnants remain a theoretically viable dark matter candidate, their formation via Gaussian primordial fluctuations is observationally excluded by LIGO. Survival of the hypothesis depends entirely on the existence of strong non-Gaussianities in the early universe.