Curl up with a good B\mathbf B: Detecting ultralight dark matter with differential magnetometry

Ultralight dark matter (such as kinetically mixed dark-photon dark matter or axionlike dark matter) can source an oscillating magnetic-field signal at the Earth's surface, which can be measured by a synchronized array of ground-based magnetometers. The global signal of ultralight dark matter can be robustly predicted for low masses, when the wavelength of the dark matter is larger than the radius of the Earth, $\lambda_\mathrm{DM}\gg R$. However, at higher masses, environmental effects, such as the Schumann resonances, can become relevant, making the global magnetic-field signal $\mathbf B$ difficult to reliably model. In this work, we show that $\nabla\times\mathbf B$ is robust to global environmental details, and instead only depends on the local dark matter amplitude. We therefore propose to measure the local curl of the magnetic field at the Earth's surface, as a means for detecting ultralight dark matter with $\lambda_\mathrm{DM}\lesssim R$. As this measurement requires vertical gradients, it can be done near a hill/mountain. Our measurement scheme not only allows for a robust prediction, but also acts as a background rejection scheme for external noise sources. We show that our technique can be the most sensitive terrestrial probe of dark-photon dark matter for frequencies $10\,\mathrm{Hz}\leq f_{A'}\leq1\,\mathrm{kHz}$ (corresponding to masses $4\times10^{-14}\,\mathrm{eV}\leq m_{A'}\leq4\times10^{-12}\,\mathrm{eV}$). It can also achieve sensitivities to axionlike dark matter comparabe to the CAST helioscope, in the same frequency range.Comment: 16 pages, 3 figure

Maglev for Dark Matter: Dark-photon and axion dark matter sensing with levitated superconductors

Ultraprecise mechanical sensors offer an exciting avenue for testing new physics. While many of these sensors are tailored to detect inertial forces, magnetically levitated (Maglev) systems are particularly interesting, in that they are also sensitive to electromagnetic forces. In this work, we propose the use of magnetically levitated superconductors to detect dark-photon and axion dark matter through their couplings to electromagnetism. Several existing laboratory experiments search for these dark-matter candidates at high frequencies, but few are sensitive to frequencies below $\mathrm{1\,kHz}$ (corresponding to dark-matter masses $m_\mathrm{DM}\lesssim10^{-12}\,\mathrm{eV}$). As a mechanical resonator, magnetically levitated superconductors are sensitive to lower frequencies, and so can probe parameter space currently unexplored by laboratory experiments. Dark-photon and axion dark matter can source an oscillating magnetic field that drives the motion of a magnetically levitated superconductor. This motion is resonantly enhanced when the dark matter Compton frequency matches the levitated superconductor's trapping frequency. We outline the necessary modifications to make magnetically levitated superconductors sensitive to dark matter, including specifications for both broadband and resonant schemes. We show that in the $\mathrm{Hz}\lesssim f_\mathrm{DM}\lesssim\mathrm{kHz}$ frequency range our technique can achieve the leading sensitivity amongst laboratory probes of both dark-photon and axion dark matter.Comment: 24 pages, 7 figure

Generalizations of the Szemerédi–Trotter Theorem

We generalize the Szemerédi–Trotter incidence theorem, to bound the number of complete flags in higher dimensions. Specifically, for each i=0,1,…,d−1, we are given a finite set S[subscript i] of i-flats in ℝ[superscript d] or in ℂ[superscript d], and a (complete) flag is a tuple (f[subscript 0],f[subscript 1],…,f[subscript d−1]), where f[subscript i]∈S[subscript i] for each i and f[subscript i]⊂f[subscript i+1] for each i=0,1,…,d−2. Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in ℝ[superscript 3] such that among the lines incident to a point, at most O(1) of them can be coplanar, (ii) incidences with Legendrian lines in ℝ[superscript 3], a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in ℝ[superscript 3] (involving points, lines, and planes), where no given line can contain too many points or lie on too many planes. The bound that we obtain in (iii) is nearly tight in the worst case. Finally, we explore a group theoretic interpretation of flags, a generalized version of which leads us to new incidence problems

Generalizations of the Szemerédi-Trotter Theorem

Abstract In this paper, we generalize the Szemerédi-Trotter theorem, a fundamental result of incidence geometry in the plane, to flags in higher dimensions. In particular, we employ a stronger version of the polynomial cell decomposition technique, which has recently shown to be a powerful tool, to generalize the Szemerédi-Trotter Theorem to an upper bound for the number of incidences of complete flags in R n (i.e. amongst sets of points, lines, planes, etc.). We also consider variants of this problem in three dimensions, such as the incidences of points and light-like lines, as well as the incidences of points, lines, and planes, where the number of points and planes on each line is restricted. Finally, we explore a group theoretic interpretation of flags, which leads us to new incidence problems