  The observed behavior of a short gamma-ray burst (sGRB) afterglow lightcurve
can reveal the angular structure of the relativistic jet and constrain the
observer's viewing angle $\theta_\textrm{obs}$. Regardless of viewing angle,
the afterglow emission is produced by the interaction between the relativistic
jet and its surrounding environment. However, the observed deceleration time of
the jet, and, therefore, the time of the afterglow peak, depends on the
observer's viewing angle. A larger viewing angle leads to a later peak of the
afterglow and a lower flux at peak. We use the the earliest X-ray afterglow
detections of 58 cosmological sGRBs detected with the Neil Gehrels Swift
Observatory X-ray Telescope to set an upper limit on the ratio of the viewing
angle $\theta_\textrm{obs}$ to the jet's half-opening angle
$\theta_\textrm{c}$. We adopt a power-law angular jet structure in both energy
$E(\theta)\propto\theta^{-a}$ and Lorentz factor
$\Gamma(\theta)\propto\theta^{-b}$ beyond the core. For this structured jet
scenario we find that either sGRBs are viewed within
$\theta_\textrm{obs}/\theta_\textrm{c}<1$ or the initial Lorentz factor of
material in their jet's core is extremely high ($\Gamma_0>500$). If we consider
a tophat jet structure, we constrain 90% of our sample to be viewed within
$\theta_\textrm{obs}/\theta_\textrm{c}<1.06$ and 1.15 for our canonical and
conservative afterglow scenarios. For a subset of events with measurements of
the jet break, under the assumption that they on-axis we can constrain
$\Gamma_0\theta_\textrm{c}\gtrsim 30$. This confirmation that cosmological
sGRBs are viewed either on-axis or very close to their jet's core has
significant implications for the nature of the prompt gamma-ray production
mechanism and for the rate of future sGRB detections coincident with
gravitational waves (GWs), implying that they are extremely rare.
