Spectroscopy as a Probe of Spacetime Foam

Abstract: Quantum-gravitational spacetime foam varying in space and in time imprints, respectively, a transverse phase decoherence and an achromatic, stochastic arrival-time jitter on light from distant sources. We show that the jitter modifies the first-order temporal coherence $g<sup>{(1)}(τ)$ and hence the observed spectrum, providing a spectroscopic probe of the temporal variation, complementary to existing imaging-channel constraints on the spatial variation. In the dynamical-foam limit, where the foam decorrelates within the light's coherence time, the spectrum splits into a coherent line core suppressed by $e<sup>{-σ_φ<sup>2}$, with $σ<em>φ<sup>2$ the variance of the foam-induced phase jitter, and an incoherent pedestal, yielding a spectral shape-distortion bound on the foam exponent $α$. Three probes follow: the outright survival of broadband flux, the fractional width of a spectral line, and a chromatic deficit across a broadband continuum. Applied to the MeV emission lines of GRB\,221009A we obtain $α\gtrsim 0.63$, a weak, lever-arm-dominated floor. For the broadband CMB the foam induces an $e^{-σ</em>φ<sup>2(ν)}$ deficit ($σ<em>φ² \propto ν<sup>2$); a fit to the COBE/FIRAS monopole spectrum reaches $α\gtrsim 0.52$ ($95\%$ CL), and next-generation spectrometers reach only $α\gtrsim 0.54$--$0.55$ (forecast), limited by the foam--temperature degeneracy. The pedestal-independent line-width and CMB bounds are floors below the holographic value $α= 2/3$, consistent with and excluding no part of the imaging window reported for GRB\,221009A; only the conditional TeV survival bound ($α\gtrsim0.74$), which holds only for near-Planck-refresh foam ($τ</em>{\rm foam}\lesssim4\times10<sup>{-28}$\,s), would exclude it --- and, with it, the window.