  A time-dependent analysis of the $B_s^0 \to \phi\gamma$ decay rate is
performed to determine the $CP$-violating observables $S_{\phi\gamma}$ and
$C_{\phi\gamma}$, and the mixing-induced observable
$\mathcal{A}^{\Delta}_{\phi\gamma}$. The measurement is based on a sample of
$pp$ collision data recorded with the LHCb detector, corresponding to an
integrated luminosity of 3 fb$^{-1}$ at center-of-mass energies of 7 and 8 TeV.
The measured values are \begin{align*} S_{\phi\gamma} &= 0.43 \pm 0.30 \pm
0.11, \\ C_{\phi\gamma} &= 0.11 \pm 0.29 \pm 0.11, \\
\mathcal{A}^{\Delta}_{\phi\gamma} &= -0.67 \, ^{+0.37}_{-0.41} \pm 0.17,
\end{align*} where the first uncertainty is statistical and the second
systematic. This is the first measurement of the observables $S$ and $C$ in
radiative $B_s^0$ decays. The results are consistent with the Standard Model
predictions.
