Limits on Spacetime Foam

Plausibly spacetime is "foamy" on small distance scales, due to quantum fluctuations. We elaborate on the proposal to detect spacetime foam by looking for seeing disks in the images of distant quasars and AGNs. This is a null test in the sense that the continued presence of unresolved "point" sources at the milli-arc second level in samples of distant compact sources puts severe constraints on theories of quantized spacetime foam at the Planckian level. We discuss the geometry of foamy spacetime, and the appropriate distance measure for calculating the expected angular broadening. We then deal with recent data and the constraints they put on spacetime foam models. While time lags from distant pulsed sources such as GRBs have been posited as a possible test of spacetime foam models, we demonstrate that the time-lag effect is rather smaller than has been calculated, due to the equal probability of positive and negative fluctuations in the speed of light inherent in such models. Thus far, images of high-redshift quasars from the Hubble Ultra-Deep Field (UDF) provide the most stringent test of spacetime foam theories. While random walk models ($\alpha = 1/2$) have already been ruled out, the holographic ($\alpha=2/3$) model remains viable. Here $\alpha \sim 1$ parametrizes the different spacetime foam models according to which the fluctuation of a distance $l$ is given by $\sim l^{1 - \alpha} l_P^{\alpha}$ with $l_P$ being the Planck length. Indeed, we see a slight wavelength-dependent blurring in the UDF images selected for this study. Using existing data in the {\it Hubble Space Telescope (HST)} archive we find it is impossible to rule out the $\alpha=2/3$ model, but exclude all models with $\alpha<0.65$. By comparison, current GRB time lag observations only exclude models with $\alpha<0.3$.