Observational Signatures of Planck-Mass Dark Matter

The canvas model predicts that dark matter consists of Planck-mass black hole remnants formed at the Planck epoch. These remnants have mass M_{\text{rem}} \approx M_P \approx 2 \times 10^{-8} kg, size r_{\text{rem}} \sim \ell_P \approx 1.6 \times 10^{-35} m, and interact only gravitationally with a cross-section \sigma \sim 10^{-70} m². This paper derives the complete observational signature of this dark matter candidate and compares it with current experimental constraints and future detection prospects. What this paper provides: · A proof that direct detection is impossible. The interaction cross-section is 23 orders of magnitude below current WIMP search sensitivity. The event rate in a ton-scale detector is \sim 10^{-58} s⁻¹—approximately one event per 10^{50} years. The canvas model predicts that continued null results from XENONnT, LZ, PandaX, and future detectors are confirmations of the model, not failures.· A proof that indirect detection is impossible. Remnants do not annihilate (they are stable black holes, not particle-antiparticle pairs) and do not decay (there is no lower-mass state). No gamma-ray, cosmic-ray, or neutrino signals from dark matter annihilation are expected. Null results from galactic center, dwarf galaxy, and solar neutrino searches are predictions.· A derivation of gravitational wave echoes from remnant mergers. When two remnants merge, they form a 2M_P black hole that evaporates via Hawking radiation back to M_P in approximately one Planck time. The evaporation burst produces a characteristic gravitational wave echo with frequency f \sim 1/t_P \sim 10^{43} Hz and strain h \sim 10^{-60} at cosmological distances—far below current detector sensitivity.· A derivation of the stochastic gravitational wave background from mergers throughout cosmic history, with energy density \Omega_{\text{GW}}(f) \sim \alpha_0 (f/f_P)^{2/3}. At accessible frequencies (10^{-9} to 10^3 Hz), \Omega_{\text{GW}} \sim 10^{-30}—far below pulsar timing array and LIGO sensitivity.· An analysis of microlensing constraints. The Einstein radius for a Planck-mass lens is \sim 10^{-7} m at galactic distances—ten million times smaller than the wavelength of visible light. Microlensing by individual remnants is completely unobservable. Remnants are far below the mass range constrained by OGLE, MACHO, and HSC surveys.· A demonstration that structure formation remains standard. Remnants are non-relativistic at all epochs after the Planck time, with free-streaming length \sim \ell_P. They behave as perfectly cold dark matter, producing no suppression of small-scale structure and matching all \LambdaCDM predictions.· A clear falsifiability criterion. The model is falsified if any of the following occur: a statistically significant WIMP signal in direct detection, a gamma-ray line or excess from dark matter annihilation, an axion signal in a haloscope, an X-ray line from sterile neutrino decay, or missing energy signatures at colliders. Continued null results are consistent with the model and explain why decades of searches have found nothing. Why this matters: Planck-mass dark matter is the most economical dark matter candidate. It requires no new particles, no new symmetries, and no new physics beyond the discrete structure of spacetime at the Planck scale. It predicts null results for all ongoing dark matter searches, consistent with current data. It is falsifiable: a single positive detection of a WIMP, axion, or sterile neutrino would rule it out. The most promising signature—gravitational wave echoes from mergers—is currently inaccessible but represents a definitive test of the model should technology ever reach Planckian frequencies. Until then, the model stands as the simplest explanation for the dark matter puzzle. Keywords: dark matter, Planck mass, black hole remnants, Planck-scale dark matter, gravitational wave echoes, direct detection, indirect detection, microlensing, structure formation, falsifiability, canvas model坎瓦思模型（canvas model）预言，暗物质由形成于普朗克纪元（Planck epoch）的普朗克质量黑洞遗迹构成。这类遗迹的质量 $M_{ ext{rem}} approx M_P approx 2 imes 10^{-8} ext{ kg}$，尺寸 $r_{ ext{rem}} sim ell_P approx 1.6 imes 10^{-35} ext{ m}$，仅通过引力相互作用，相互作用截面 $sigma sim 10^{-70} ext{ m}^2$。 本研究推导了这类暗物质候选体的完整观测特征，并将其与当前实验约束及未来探测前景进行了对比。 本文的主要贡献包括： · 证明直接探测不可行。其相互作用截面比当前弱相互作用大质量粒子（WIMP, Weakly Interacting Massive Particle）的探测灵敏度低23个数量级。吨级探测器内的事件率 $sim 10^{-58} ext{ s}^{-1}$——约每 $10^{50}$ 年仅发生一次事件。坎瓦思模型预言，XENONnT、LZ、PandaX等现有探测器以及未来探测装置持续给出的零结果，正是对该模型的佐证，而非探测失败。 · 证明间接探测不可行。遗迹不会发生湮灭（它们是稳定的黑洞，而非粒子-反粒子对），也不会发生衰变（不存在质量更低的能态）。因此不会产生暗物质湮灭带来的伽马射线、宇宙线或中微子信号。银心、矮星系以及太阳中微子搜寻得到的零结果，均为该模型的预言。 · 推导了遗迹并合产生的引力波回波信号。当两个遗迹发生并合时，会形成质量为 $2M_P$ 的黑洞，该黑洞会通过霍金辐射在约1个普朗克时间内蒸发回 $M_P$ 的质量。此次蒸发爆发会产生特征引力波回波，其频率 $f sim 1/t_P sim 10^{43} ext{ Hz}$，在宇宙学距离下的应变 $h sim 10^{-60}$——远低于当前探测器的灵敏度阈值。 · 推导了整个宇宙历史中并合事件产生的随机引力波背景，其能量密度 $Omega_{ ext{GW}}(f) sim alpha_0 (f/f_P)^{2/3}$。在可探测频段（$10^{-9}$ 至 $10^3 ext{ Hz}$）内，$Omega_{ ext{GW}} sim 10^{-30}$——远低于脉冲星计时阵列（pulsar timing array）以及激光干涉引力波天文台（LIGO, Laser Interferometer Gravitational-Wave Observatory）的探测灵敏度。 · 分析了微引力透镜约束。对于普朗克质量的透镜天体，其在银河距离处的爱因斯坦半径 $sim 10^{-7} ext{ m}$——比可见光波长小1000万倍。单个遗迹产生的微引力透镜效应完全无法被观测到，且该类遗迹的质量远低于光学引力透镜实验（OGLE, Optical Gravitational Lensing Experiment）、大质量致密晕天体巡天（MACHO, Massive Compact Halo Object Survey）以及HSC巡天（HSC）所约束的质量范围。 · 证明结构形成过程符合标准模型。遗迹在普朗克纪元之后的所有宇宙时期均为非相对论性，其自由程 $sim ell_P$。它们表现为完全的冷暗物质，不会对小尺度结构产生抑制，且符合所有Λ冷暗物质（ΛCDM, Lambda Cold Dark Matter）模型的预言。 · 明确了该模型的可证伪性判据。若出现以下任意一种情况，则该模型被证伪：直接探测中出现统计显著的WIMP信号、暗物质湮灭产生的伽马射线谱线或过剩信号、轴子（axion）探测器中出现轴子信号、无菌中微子衰变产生的X射线谱线，或是对撞机中出现缺失能量信号。持续的零结果与该模型相符，这也解释了为何数十年来的暗物质搜寻均未获得阳性结果。 研究意义： 普朗克质量暗物质是最经济的暗物质候选体。它无需引入新粒子、新对称性，也无需超出普朗克尺度时空离散结构的新物理。该模型预言所有正在进行的暗物质搜寻均会得到零结果，这与当前实验数据相符。该模型具备可证伪性：只要探测到WIMP、轴子或无菌中微子中的任意一种阳性信号，即可排除该模型。最具前景的特征信号——并合产生的引力波回波——目前仍无法被探测，但当技术发展至普朗克频段时，该信号将成为验证该模型的决定性手段。在此之前，该模型仍是解释暗物质谜题的最简方案。 关键词：暗物质，普朗克质量，黑洞遗迹，普朗克尺度暗物质，引力波回波，直接探测，间接探测，微引力透镜，结构形成，可证伪性，坎瓦思模型（canvas model）