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arxiv: 2605.12948 · v1 · pith:XOVA52VInew · submitted 2026-05-13 · ✦ hep-ph · astro-ph.CO· gr-qc· hep-th

Secondary Gravitational Wave Signatures from 5D Rotating Primordial Black Holes in the Dark Dimension

Waqas Ahmed , George K. Leontaris This is my paper

Pith reviewed 2026-05-14 18:59 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COgr-qchep-th
keywords primordial black holesextra dimensionsgravitational wavesdark matterHawking evaporationmemory burdenstochastic backgroundDark Dimension
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The pith

Rotating five-dimensional primordial black holes produce a stochastic gravitational wave background detectable by LISA and DECIGO that could confirm a micron-scale extra dimension, PBH dark matter, and memory burden suppression.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper calculates how five-dimensional rotating primordial black holes stabilized by a micron-scale extra dimension and the memory burden effect can survive as dark matter over a wide mass range. Curvature perturbations that formed these black holes also induce a second-order stochastic gravitational wave background whose present-day energy density depends on the black hole mass window, the dark matter fraction, and the memory burden exponent. The resulting spectrum peaks between nanohertz and hertz frequencies and falls within the reach of planned detectors while respecting existing CMB bounds. Fisher forecasts indicate that a detection would simultaneously constrain the PBH mass, dark matter fraction, spectral width, and memory burden parameter to percent-level precision.

Core claim

In the Dark Dimension framework a micron-scale extra dimension suppresses Hawking evaporation for five-dimensional rotating primordial black holes, while the memory burden effect further reduces the evaporation rate by a factor S^{-p} with p=2. This combination allows initial masses between 10^{10} g and 10^{21} g to persist to the present day as dark matter. The same curvature perturbations that seed these black holes generate, through second-order scalar-induced effects, a stochastic gravitational wave background with Omega_GW h^2 peaking in the nHz-Hz band for a log-normal primordial spectrum with sigma=1 and f_PBH=1; the amplitude and shape are set by the extended PBH lifetime and remain

What carries the argument

The second-order scalar-induced gravitational wave spectrum computed from a log-normal primordial curvature power spectrum, modulated by the prolonged evaporation lifetime of 5D rotating black holes under the memory burden effect with exponent p=2 in the Dark Dimension geometry.

If this is right

  • The predicted gravitational wave background lies inside the sensitivity curves of LISA and DECIGO/BBO for black hole masses between 10^{10} g and 10^{21} g.
  • Future observations can extract the PBH mass, dark matter fraction, spectral width, and memory burden exponent to percent-level accuracy via Fisher matrix analysis.
  • The signal amplitude remains compatible with current CMB spectral distortion limits across the allowed parameter space.
  • A positive detection would simultaneously indicate the presence of a micron-scale extra dimension, primordial black hole dark matter, and memory burden suppression of evaporation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same framework could be applied to non-rotating five-dimensional black holes or to different values of the extra-dimension radius to map how the gravitational wave spectrum shifts.
  • Constraints from a detection could be fed back to refine the allowed range of the memory burden exponent or the precise size of the dark dimension.
  • This calculation connects the formation of primordial black holes directly to testable predictions in extra-dimensional quantum gravity models without requiring additional free parameters beyond those already fixed by the dark matter requirement.

Load-bearing premise

The assumption that the primordial power spectrum is log-normal with fixed width sigma=1, that the dark matter fraction equals one, and that the memory burden exponent equals two for five-dimensional rotating black holes, all of which fix the amplitude and frequency location of the gravitational wave signal.

What would settle it

A null result or a spectral shape inconsistent with the predicted peak frequency and amplitude in the nHz-Hz band from LISA or DECIGO observations would rule out the specific combination of Dark Dimension radius, PBH mass window, and memory burden exponent p=2 used in the calculation.

Figures

Figures reproduced from arXiv: 2605.12948 by George K. Leontaris, Waqas Ahmed.

Figure 1
Figure 1. Figure 1: FIG. 1: Ratio of the five-dimensional Schwarzschild radius [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: illustrates the coupled evolution of mass and angular momentum during the spin-down phase. Curves are obtained by numerically inverting the implicit relation for different η and initial spins L0. Higher initial spin results in a longer spin-down phase and larger mass loss (typically 40%–60% of the initial mass is radiated away). The spin-down timescale is τsp ≈ 5.2 × 10−15(Mi g)2 years; for Mi = 5 × 1011 g… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: PBH lifetime as a function of initial mass for different evaporation scenarios, computed using the equation [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: primordial curvature power spectrum [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Present-day scalar-induced gravitational wave spectra Ω [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Fisher corner plots for LISA, DECIGO, and BBO. Diagonal panels show marginalised posteriors (vertical [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
read the original abstract

We investigate five-dimensional rotating primordial black holes (PBHs) as dark matter candidates within the Dark Dimension (DD) scenario motivated by the Swampland Program. In this framework, a micron-scale extra dimension suppresses Hawking evaporation, allowing PBHs with initial masses \(M \gtrsim 10^{10}\,\mathrm{g}\) to survive to the present epoch. Moreover, the memory burden effect, a quantum-gravitational suppression of the evaporation rate by \(S^{-p}\), significantly prolongs PBH lifetimes and enlarges the allowed parameter space. We compute the evaporation dynamics for rotating 5D PBHs, derive the enhanced lifetime for \(p=2\), and establish the dark matter window \(10^{10}\,\mathrm{g} \lesssim M \lesssim 10^{21}\,\mathrm{g}\). The curvature perturbations responsible for PBH formation also generate a stochastic gravitational wave background through second-order scalar-induced effects. Assuming a log-normal primordial power spectrum with \(\sigma=1\) and \(f_{\mathrm{PBH}}=1\), we calculate the present-day energy density \(\Omega_{\mathrm{GW}}h^2\) across the Dark Dimension window. The predicted signals peak at frequencies from nHz to Hz, within the sensitivity ranges of LISA and DECIGO/BBO, while remaining consistent with current CMB spectral distortion bounds. Fisher forecasts show that future observatories can constrain the PBH mass, dark matter fraction, spectral width, and memory burden exponent with percent-level precision. A detection of the predicted gravitational wave background would provide simultaneous evidence for a micron-sized extra dimension, PBH dark matter, and the memory burden effect, offering a decisive test of quantum gravity and extra-dimensional physics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript investigates five-dimensional rotating primordial black holes (PBHs) as dark matter candidates within the Dark Dimension scenario. It incorporates the memory burden effect (S^{-p} suppression with p=2) to prolong evaporation lifetimes, establishes a viable DM mass window 10^{10} g ≲ M ≲ 10^{21} g, and computes the stochastic gravitational wave background from second-order scalar-induced effects. Assuming a log-normal primordial power spectrum with fixed σ=1 and f_PBH=1, the present-day Ω_GW h² is calculated across the allowed window, with peaks in the nHz–Hz range accessible to LISA and DECIGO/BBO. Fisher forecasts are presented for percent-level constraints on PBH mass, f_PBH, σ, and p; a detection is claimed to simultaneously confirm the micron-scale extra dimension, PBH dark matter, and memory burden effect.

Significance. If the evaporation dynamics and GW spectrum calculations hold, the work links Swampland-motivated extra-dimensional physics to a concrete, multi-messenger observable. It provides a potential falsifiable test that could constrain the Dark Dimension radius and memory-burden exponent through the amplitude, frequency, and shape of the induced GW background, while remaining consistent with existing CMB bounds.

major comments (3)
  1. Abstract: The computation of Ω_GW h² assumes f_PBH=1 and log-normal width σ=1 to fix the perturbation amplitude A; the resulting signal strength and peak location are therefore conditional on these inputs rather than an independent prediction of the 5D rotating PBH framework. This directly undermines the headline claim that a detection would provide simultaneous evidence for PBH dark matter via f_PBH=1.
  2. Abstract: The memory-burden exponent is fixed at p=2 without derivation from the 5D Swampland setup or 5D Kerr geometry; varying p changes the evaporation rate, the allowed mass window, and the frequency range of the GW peak, so the quoted LISA/DECIGO sensitivity window is not robust to this choice.
  3. Evaporation dynamics and GW spectrum sections: The derivations of the enhanced lifetime for rotating 5D PBHs, the numerical evaluation of the scalar-induced spectrum, and the Fisher forecasts cannot be verified from the provided details; these steps are load-bearing for the central claim that the signal offers a decisive test of quantum gravity and extra dimensions.
minor comments (1)
  1. Abstract: The statement that the signals remain consistent with CMB spectral distortion bounds does not specify which bounds (e.g., μ-distortion limits) or the quantitative margin by which they are satisfied.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below with clarifications on our assumptions and enhancements to the derivations. Revisions have been incorporated to improve transparency and robustness while preserving the core results.

read point-by-point responses

  1. Referee: Abstract: The computation of Ω_GW h² assumes f_PBH=1 and log-normal width σ=1 to fix the perturbation amplitude A; the resulting signal strength and peak location are therefore conditional on these inputs rather than an independent prediction of the 5D rotating PBH framework. This directly undermines the headline claim that a detection would provide simultaneous evidence for PBH dark matter via f_PBH=1.

    Authors: We agree that the GW amplitude and peak frequency are computed for the benchmark values f_PBH=1 and σ=1, which fix the perturbation amplitude A and maximize the signal within the allowed mass window. These choices are explicitly stated as benchmarks in the manuscript to represent the case of PBH dark matter. The mass window itself follows from the 5D evaporation dynamics independently of the power spectrum details. We have revised the abstract to clarify that the quoted signal and sensitivity claims apply to this benchmark scenario, and that a detection would support the framework under these assumptions (with lower f_PBH scaling down the amplitude proportionally). This tones down the simultaneous-evidence phrasing while retaining the multi-messenger test aspect. revision: partial

  2. Referee: Abstract: The memory-burden exponent is fixed at p=2 without derivation from the 5D Swampland setup or 5D Kerr geometry; varying p changes the evaporation rate, the allowed mass window, and the frequency range of the GW peak, so the quoted LISA/DECIGO sensitivity window is not robust to this choice.

    Authors: The value p=2 is adopted as a representative choice from the memory-burden literature, where it corresponds to a moderate quantum-gravitational suppression. We acknowledge that it is not uniquely derived from the 5D Swampland or Kerr geometry in the present work. In the revision we have added a dedicated paragraph in Section 2 and an appendix figure showing the dependence of the lifetime, mass window, and GW peak frequency on p (for p=1,2,3). The abstract now specifies that the LISA/DECIGO ranges refer to p=2, with the general framework permitting shifts in the observable window for other p values consistent with the Swampland motivation. revision: yes

  3. Referee: Evaporation dynamics and GW spectrum sections: The derivations of the enhanced lifetime for rotating 5D PBHs, the numerical evaluation of the scalar-induced spectrum, and the Fisher forecasts cannot be verified from the provided details; these steps are load-bearing for the central claim that the signal offers a decisive test of quantum gravity and extra dimensions.

    Authors: We have expanded the relevant sections and added a new Appendix C containing the full step-by-step derivation of the 5D Kerr evaporation rate (including the angular momentum dependence and the S^{-p} memory-burden factor), the explicit integral expression for the scalar-induced Ω_GW spectrum with the log-normal power spectrum, and the Fisher matrix construction with all parameter derivatives and covariance terms. Numerical quadrature methods, integration limits, and convergence checks are now specified. These additions render the calculations fully verifiable while leaving the central results unchanged. revision: yes

Circularity Check

2 steps flagged

GW amplitude and frequency range are set by fixed σ=1, f_PBH=1 and p=2 with no derivation from the 5D Swampland setup

specific steps
  1. fitted input called prediction [Abstract]
    "Assuming a log-normal primordial power spectrum with σ=1 and f_PBH=1, we calculate the present-day energy density Ω_GW h² across the Dark Dimension window."

    The Ω_GW h² spectrum is obtained by direct substitution of the assumed log-normal form (σ=1) and f_PBH=1 into the standard second-order scalar-induced formula; the resulting amplitude and peak location are therefore fixed by these input choices rather than predicted independently from the 5D metric or memory-burden dynamics.

  2. fitted input called prediction [Abstract]
    "the memory burden effect, a quantum-gravitational suppression of the evaporation rate by S^{-p}, significantly prolongs PBH lifetimes... derive the enhanced lifetime for p=2"

    The lifetime enhancement and resulting DM window are computed after selecting the specific exponent p=2; the allowed mass range 10^{10} g ≲ M ≲ 10^{21} g and the GW frequency window are therefore set by this choice rather than emerging from the 5D Kerr geometry alone.

full rationale

The paper computes Ω_GW h² only after inserting a log-normal P_ζ(k) with σ=1 and f_PBH=1 to fix the perturbation amplitude A, then applies the S^{-p} suppression with chosen p=2 to the 5D Kerr evaporation rate. These three numbers are not derived from the 5D geometry or Swampland constraints; they are chosen by hand. Changing any one moves the predicted peak outside the LISA/DECIGO window or below sensitivity, so the claimed 'decisive test' reduces to the specific numerical inputs rather than to the 5D rotating PBH framework itself.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Dark Dimension scenario (micron extra dimension suppressing evaporation) and the memory burden effect, both introduced as motivated assumptions rather than derived; free parameters include the memory burden exponent, spectral width, and PBH fraction.

free parameters (3)
  • memory burden exponent p = 2
    Set to 2 to significantly prolong PBH lifetimes in the 5D setup
  • log-normal spectral width sigma = 1
    Fixed at 1 for the primordial power spectrum used to compute GW background
  • PBH dark matter fraction f_PBH = 1
    Set to 1 to saturate the dark matter density in the calculated window
axioms (2)
  • domain assumption Dark Dimension scenario with micron-scale extra dimension that suppresses Hawking evaporation for PBHs
    Motivated by the Swampland Program; invoked to allow M ≳ 10^10 g PBHs to survive to today
  • domain assumption Memory burden effect providing quantum-gravitational suppression of evaporation rate by S^{-p}
    Applied with p=2 to enlarge the allowed PBH mass window

pith-pipeline@v0.9.0 · 5624 in / 1601 out tokens · 43670 ms · 2026-05-14T18:59:11.176000+00:00 · methodology

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