⚛️ general relativity

Different effects of the Lorentz and Gaussian bump functions on the formation of primordial black holes and secondary gravitational waves

This paper demonstrates that, when applied to the Starobinsky inflation potential with identical parameters, Lorentzian bump functions are more effective than Gaussian ones at amplifying the curvature power spectrum, thereby generating a greater abundance of primordial black holes and stronger secondary gravitational waves.

Original authors: Wei Yang, Yu-Xuan Kang, Arshad Ali, Tao-Tao Sui, Chen-Hao Wu, Ya-Peng Hu

Published 2026-01-23

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CC BY 4.0

Original authors: Wei Yang, Yu-Xuan Kang, Arshad Ali, Tao-Tao Sui, Chen-Hao Wu, Ya-Peng Hu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the very early universe as a giant, smooth trampoline. In the standard story of how the universe began (called "inflation"), this trampoline stretches out smoothly and evenly. But sometimes, scientists think there might have been a tiny, localized bump on that trampoline. This bump isn't a physical object; it's a slight change in the energy rules that governed the universe's expansion.

This paper is like a scientific taste test. The researchers wanted to see: Does it matter what shape that tiny bump has?

They tested two famous shapes:

The Gaussian Bump: Think of this as a perfect, symmetrical hill, like a classic sand dune or a bell curve. It rises sharply and falls off very quickly.
The Lorentz Bump: Think of this as a wider, flatter hill with "fat tails." It rises similarly but stays high for longer and tapers off much more slowly, like a gentle, rolling plateau.

Here is what they found, using simple analogies:

1. The "Slow-Down" Effect

When the universe expanded over these bumps, the "speed" of expansion changed.

The Gaussian hill was like a steep ramp. The universe rolled over it quickly.
The Lorentz hill was like a long, flat plateau. The universe spent much more time "stuck" or moving very slowly over this wider area.

2. Making Primordial Black Holes (The "Snowball" Effect)

Because the universe slowed down so much more on the Lorentz hill, it created massive ripples in the fabric of space-time. Imagine throwing a stone into a pond; a Lorentz bump creates a huge, crashing wave, while a Gaussian bump creates just a small ripple.

These huge waves were strong enough to crush matter together, forming Primordial Black Holes (PBHs)—tiny, ancient black holes that formed right after the Big Bang.

The Result: The Lorentz bump was a "black hole factory." It produced a huge abundance of these black holes.
The Gaussian bump: It barely made any. The ripples were too weak to crush matter into black holes.

The paper suggests that if we find these ancient black holes in the universe today (perhaps explaining the dark matter that holds galaxies together), it might be because the universe had a "Lorentz-style" bump, not a Gaussian one.

3. The "Echo" (Gravitational Waves)

When those huge waves crashed to form black holes, they also created a secondary effect: Gravitational Waves. Think of this as the "echo" or the "rumble" that follows a thunderclap.

The Lorentz bump created a very loud, energetic rumble (high energy density). This signal is strong enough that future space telescopes (like LISA or TianQin) might actually hear it.
The Gaussian bump created a whisper that is likely too quiet to detect.

The Bottom Line

The researchers didn't invent new physics; they just compared two different mathematical shapes for the same event. They found that the Lorentz shape is much more effective at:

Creating a "fat" plateau that slows down the universe's expansion.
Generating enough ripples to form a large number of ancient black holes.
Creating a loud enough gravitational wave signal to be detected by future instruments.

In short: If the universe had a "wide, flat" bump (Lorentz), we would expect to see many ancient black holes and hear their gravitational echoes. If it had a "sharp, narrow" bump (Gaussian), we would see very few of either. This helps scientists decide which mathematical model to use when trying to explain what we might observe in the future.

Technical Summary: Different Effects of Lorentz and Gaussian Bump Functions on Primordial Black Holes and Secondary Gravitational Waves

Problem Statement
Primordial Black Holes (PBHs) are viable candidates for cold dark matter (DM) and potential sources of gravitational wave (GW) events. Their formation is often attributed to the amplification of scalar perturbations during inflation, which can be achieved by introducing a local "bump" to the base inflation potential, $V(\phi) = V_b(\phi)(1 + f(\phi))$ . While various bump functions (Gaussian, hyperbolic, Lorentz) have been utilized in literature, there is no underlying physical principle dictating the specific choice of $f(\phi)$ . Consequently, it remains unclear how the specific functional form of the bump influences the resulting curvature power spectrum, the abundance of PBHs, and the induced secondary gravitational waves (SIGWs). This paper addresses the lack of comparative analysis between widely used bump functions, specifically the Lorentz and Gaussian forms, under identical base conditions.

Methodology
The authors employ the Starobinsky inflation model as the base potential $V_b(\phi)$ , chosen for its compatibility with Cosmic Microwave Background (CMB) observations. They introduce local correction bumps $f(\phi)$ in two forms:

Lorentz Bump: $f_L(\phi) = \frac{b}{1 + (\frac{\phi-\phi_0}{c})^2}$
Gaussian Bump: $f_G(\phi) = b e^{-\frac{(\phi-\phi_0)^2}{2c^2}}$

To isolate the effects of the functional shape, the authors keep the amplitude parameter ( $b$ ), peak position ( $\phi_0$ ), and width parameter ( $c$ ) identical for both cases. They numerically solve the Mukhanov-Sasaki equation to compute the scalar power spectrum $\mathcal{P}_\zeta$ . Using the Press-Schechter formalism, they calculate the PBH abundance ( $Y_{PBH}$ ) and the energy density spectrum of scalar-induced gravitational waves ( $\Omega_{GW}$ ). Three distinct parameter sets (I, II, III) are tested to verify the generality of the findings.

Key Contributions and Results

Enhancement of Power Spectrum: The study reveals a novel and significant difference in the ability of the two functions to enhance the power spectrum. The Lorentz bump exhibits a "fatter tail" compared to the Gaussian bump, which decays exponentially. This structural difference creates a wider and flatter plateau in the inflation potential, leading to a more extended period of ultra-slow roll. Consequently, the Lorentz case produces a peak power spectrum ( $\mathcal{P}_\zeta \sim \mathcal{O}(10^{-2})$ ) that is orders of magnitude higher than the Gaussian case ( $\mathcal{P}_\zeta \sim \mathcal{O}(10^{-7})$ ) under the same parameters.
Primordial Black Hole Abundance: Due to the significantly higher power spectrum, the Lorentz bump generates a substantial abundance of PBHs, whereas the Gaussian bump yields negligible abundance (effectively zero in the tested scenarios).
- Parameter Set I: Generates stellar-mass PBHs ( $\sim 37 M_\odot$ ), potentially relevant to LIGO binary black hole merger events.
- Parameter Set II: Produces PBHs with masses $\sim 10^{-5} M_\odot$ , which could explain OGLE ultrashort-timescale microlensing events.
- Parameter Set III: Yields PBHs with masses $\sim 10^{-12} M_\odot$ with an abundance $Y_{PBH} \approx 1$ , suggesting they could constitute all dark matter.
Scalar-Induced Gravitational Waves (SIGWs): The enhanced scalar perturbations in the Lorentz case induce a significantly higher energy density of SIGWs compared to the Gaussian case. The peak frequencies of these signals fall within the sensitivity ranges of various future and current detectors:
- Set I: Detectable by pulsar timing arrays (EPTA, PPTA, SKA) and NANOGrav.
- Set II: Detectable by SKA.
- Set III: Detectable by space-based interferometers (LISA, Taiji, TianQin, DECIGO).

Significance and Claims
The paper claims that the choice of bump function is not merely a mathematical convenience but has profound physical consequences. The Lorentz function's power-law decay allows for a more effective enhancement of perturbations compared to the Gaussian function's exponential decay. This distinction offers a potential observational handle: future experiments detecting specific PBH mass ranges or SIGW signals could distinguish between these two theoretical scenarios.

The authors modestly note that their use of the Press-Schechter formalism is a simplification and that their analysis assumes Gaussian statistics, neglecting potential non-Gaussian effects in the tails of the probability density function. They suggest that future work incorporating peak theory and non-Gaussianity, as well as investigating one-loop corrections to CMB scales, would provide a more accurate and comprehensive understanding of PBH formation mechanisms.

1. The "Slow-Down" Effect

2. Making Primordial Black Holes (The "Snowball" Effect)

3. The "Echo" (Gravitational Waves)

The Bottom Line

Technical Summary: Different Effects of Lorentz and Gaussian Bump Functions on Primordial Black Holes and Secondary Gravitational Waves

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