The minimum quantum of the acceleration ladder

In our research A dark-mass ladder in disk galaxies we wrote the acceleration of a galaxy as theproduct g2 = gN (gN + a0) and found that the dark mass is a ladder in steps of ra =√GM/a0,each step carrying the energy Eq = Mv2f with v4f = GM a0. We called the ladder a discretisationand not a quantisation, because ¯h did not enter it. Here we look for the minimum value that makes it aquantisation, and we find one mass, M∗ = (¯h4a0/G3)1/7 = 3.485·10−17 kg, at which three conditionshold together: one quantum of angular momentum in the first step, one quantum of action in the energyand time of a step, and a de Broglie wavelength equal to the step. With L = n¯ hthe ladder becomes aquantum ladder, Mn = n4/7M∗, and a body of mass M carries N = (M/M∗)7/4 quanta, from lessthan one for a proton to 4.8 · 10100 for the Milky Way. We then quantise the dark matter itself: itsquantum in a system of mass M is md = ¯h/vf ra = M∗(M∗/M)3/4, and the energy of every stepof the field is N quanta ¯hωa, ωa = vf /ra. Because md changes from galaxy to galaxy and lies belowthe Lyman-α bound on an ultralight particle for every galaxy we examined, the quantum belongs to thegravitational field: to quantise the dark matter of the law is to quantise the part of gravity set by a0.Two masses M∗ gain one radian of gravitational phase in one step. We find a0 = 1.498×10−10 m s−2on SPARC, equal to c2√Λ/2π for the Planck value of Λ; if this relation holds, M∗ is the geometricmean of the Planck mass and of the quantum of dark energy. M∗ is also the mass at which the Bohrradius of the gravitational atom equals the step of the ladder, a fourth equivalent condition. Readwith energies instead of momenta, the same product gives the law that the rotation curves select, witha0 = 1.144 × 10−10 m s−2; it changes M∗ by 4% and keeps every quantum result, but it moves a0from c2√Λ/2π (0.766 of it) towards cH0/2π (1.098), so that the link with dark energy depends onthe reading. We test the results on independent observations: five further measurements of a0, fourcosmologies, three studies of a0 at higher redshift, weak lensing to 115 kpc, and the Gaia rotation curveof the Milky Way. M∗ changes by at most 5% across them; the lensing velocities agree within 0.13 dex;the Milky Way is in tension; and the largest study at intermediate redshift finds a0 growing faster thanH(z), against the constant a0 of the relation with Λ.