Spectroscopy as a Probe of Spacetime Foam

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 $g^{(1)}(\tau )$ 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 $e^{-{\sigma }_{\phi }^2}$, with ${\sigma }_{\phi }^2$ the variance of the foam-induced phase jitter, and an incoherent pedestal, yielding a spectral shape-distortion bound on the foam exponent $\alpha$. 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 $\alpha \gtrsim 0.63$, a weak, lever-arm-dominated floor. For the broadband CMB the foam induces an $e^{-{\sigma }_{\phi }^2(\nu )}$ deficit (${\sigma }_{\phi }^2\propto {\nu }^2$); a fit to the COBE/FIRAS monopole spectrum reaches $\alpha \gtrsim 0.52$ ($95\mathrm{\%}$ CL), and next-generation spectrometers reach only $\alpha \gtrsim 0.54$–$0.55$ (forecast), limited by the foam–temperature degeneracy. The pedestal-independent line-width and CMB bounds are floors below the holographic value $\alpha =2\mathrm{/}3$, consistent with and excluding no part of the imaging window reported for GRB,221009A; only the conditional TeV survival bound ($\alpha \gtrsim 0.74$), which holds only for near-Planck-refresh foam (${\tau }_{\mathrm{f}\mathrm{o}\mathrm{a}\mathrm{m}}\lesssim 4\times {10}^{-28}$,s), would exclude it — and, with it, the window.