  Developments in the stability of modern spectrographs have led to extremely
precise instrumental radial velocity (RV) measurements. For most stars, the
detection limit of planetary companions with these instruments is expected to
be dominated by astrophysical noise sources such as starspots. Correlated
signals caused by rotationally-modulated starspots can obscure or mimic the
Doppler shifts induced by even the closest, most massive planets. This is
especially true for young, magnetically active stars where stellar activity can
cause fluctuation amplitudes of $\gtrsim$0.1 mag in brightness and $\gtrsim$100
m s$^{-1}$ in RV semi-amplitudes. Techniques that can mitigate these effects
and increase our sensitivity to young planets are critical to improving our
understanding of the evolution of planetary systems. Gaussian processes (GPs)
have been successfully employed to model and constrain activity signals in
individual cases. However, a principled approach of this technique,
specifically for the joint modeling of photometry and RVs, has not yet been
developed. In this work, we present a GP framework to simultaneously model
stellar activity signals in photometry and RVs that can be used to investigate
the relationship between both time series. Our method, inspired by the
$\textit{FF}^\prime$ framework of (Aigrain et al. 2012), models spot-driven
activity signals as the linear combinations of two independent latent GPs and
their time derivatives. We also simulate time series affected by starspots by
extending the $\texttt{starry}$ software (Luger et al. 2019) to incorporate
time evolution of stellar features. Using these synthetic datasets, we show
that our method can predict spot-driven RV variations with greater accuracy
than other GP approaches.
