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 active galactic nuclei. This is a null test in the sense that the continued presence of unresolved point sources at the milliarcsecond 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 gamma ray bursts 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 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 ultra-deep field images selected for this study. Using existing data in the 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 gamma ray burst time-lag observations only exclude models with $\alpha <0.3$.

Figure 1

The detectability of various models of foamy spacetime with existing and planned telescopes. Diagonal tracks are shown for three models of foamy spacetime, namely $\alpha =0.5$, 0.6, $2/3$, for $z=1$ and 4 (respectively the lower and upper tracks for each model) and $N=1.9$. Also shown are the observing ranges and theoretical resolution limits (i.e., PSF size) for a wide variety of telescopes, both current and planned. The tracks for each telescope represent the diffraction limit with the exception of the dotted track for Chandra, whose complicated PSF is NOT diffraction-limited (see Sec. 5). As such, they represent the detectability limit for these observations, in the case of a perfect telescope of that size. Importantly, note that in practical terms a given telescope may not be able to probe quite as far as pictured, as in the final analysis the region probed is also a function of Strehl ratio (see Sec. 6).

Figure 2

Expected Strehl ratio with redshift (left) and wavelength (right) for two spacetime foam cases: $\alpha =2/3$, $N=1.9$ (black lines); $\alpha =0.655$, $N=1.9$ (grey lines). In the left-hand figure we illustrate the change in Strehl ratio expected for a point source with varying distance, $z$, for three fixed observed wavelengths (${\lambda }_o=200$, 300 and 400 nm as marked). In the right-hand figure we show the change in Strehl ratio expected for a point source at various three fixed distances: $z=6$ (solid line), $z=4$ (dashed line) and $z=2$ (dot dashed line).

Figure 3

The four HUDF quasars studied in this paper, from left to right: 9397 $(z=1.2)$; 6732 $(z=3.2)$; 9487 $(z=4.1)$; 4120$(z=2.1)$. Descending rows show the F435W (B), F606W (V), F775W (I) and F850LP (x) images.

Figure 4

Top, from left-to-right: HUDF-QSO 6732 in F435W (B), F606W (V), F775W (i) and F850LP (z) bands. Middle: The ideal model PSF (1.2 m radius mirror with central hole of 0.35 m radius) for same filters. Bottom: Stellar PSF’s for the same filters.

Figure 5

Measured Strehl ratio $S$ for the four UDF quasars (9397: circles and solid line; 6732: up-pointing triangles and dotted line; 9487: squares and dot-dashed line; 4120: down-pointing triangles and dashed line) against theoretical Strehl ratio $S_{\varphi }$ (Eq. (16)), for various spacetime foam models. The shaded area indicates $S>S_{\varphi }$: Such observations exclude a given spacetime foam model, $(\alpha ,N)$. Crosses indicate data points from [15]. We exclude all models with $\alpha <0.65$ since we clearly see quasars that are less blurred than is predicted by such models (top). We do in fact see a small amount of blurring in our UDF quasars, consistent with the magnitude of blurring expected for $\alpha =0.655$ (middle). No degradation should be seen for $\alpha =2/3$ (bottom) for the wavelengths and distances probed so far by the HST archive ($\lambda >400\mathrm{nm}$).