  Optomechanical accelerometers promise quantum-limited readout, high detection
bandwidth, self-calibration, and radiation pressure stabilization. We present a
simple, scalable platform that enables these benefits with nano-$g$ sensitivity
at acoustic frequencies, based on a pair of vertically integrated Si$_3$N$_4$
membranes with different stiffnesses, forming an optical cavity. As a
demonstration, we integrate an ultrahigh-Q ($>10^7$), millimeter-scale
Si$_3$N$_4$ trampoline membrane above an unpatterned membrane on the same Si
chip, forming a finesse $\mathcal{F}\approx2$ cavity. Using direct
photodetection in transmission, we resolve the relative displacement of the
membranes with a shot-noise-limited imprecision of 7 fm/$\sqrt{\text{Hz}}$,
yielding a thermal-noise-limited acceleration sensitivity of 562
n$g/\sqrt{\text{Hz}}$ over a 1 kHz bandwidth centered on the fundamental
trampoline resonance (40 kHz). To illustrate the advantage of radiation
pressure stabilization, we cold damp the trampoline to an effective temperature
of 4 mK and leverage the reduced energy variance to resolve an applied
stochastic acceleration of 50 n$g/\sqrt{\text{Hz}}$ in an integration time of
minutes. In the future, we envision a small-scale array of these devices
operating in a cryostat to search for fundamental weak forces such as dark
matter.
