Search for time-dependent $CP$ violation in $D^0 \rightarrow \pi^+ \pi^- \pi^0$ decays and precision luminosity measurements at LHCb.

The results of an absolute luminosity measurement at the LHCb experiment and a search for time-dependent $CP$ violation in $D^0$ meson decays are presented in this thesis. An overview of the underlying theory relevant to time-dependent $CP$ violation in the neutral meson systems is also provided, with a particular focus on neutral charmed-mesons. The LHCb detector that recorded the data used for these measurements is also described in detail, along with its recent upgrade for Runs 3 and 4 of the LHC. An absolute determination of the luminosity at the LHCb experiment between 2016 and 2018 in $pp$ collisions at $\sqrt{s} = 13$ TeV is obtained by directly reconstructing the beam profiles from beam-gas interactions. A cross section of \begin{equation} \sigma_\mathrm{Vertex} = 58.00 \pm 0.02\, (\text{stat}) \pm 0.46\, (\text{syst})\, \text{mb} \nonumber \end{equation} for the Vertex counter is obtained, with a preliminary relative precision of 0.83 %. The largest systematic uncertainties arise from corrections to the beam-gas vertex resolution, measurement spread believed to be caused by an imperfect bunch profile description, and uncertainty on the bunch current measurements. The dominant systematic uncertainties are largely uncorrelated with the complementary van der Meer analysis. The combination of the two measurements therefore has the potential to be the most precise determination of the absolute luminosity at a hadron collider to date. A measurement of time-dependent $CP$ violation in $D^0 \rightarrow \pi^+ \pi^- \pi^0$ decays collected by the LHCb experiment in 2012 and from 2015 to 2018, corresponding to an integrated luminosity of 7.7 $\text{fb}^{-1}$, is also presented. The flavour of each $D^0$ candidate is determined from the decay $D^*(2010)^+ \rightarrow D^0\pi^+$. Detection asymmetries are corrected with a data-driven weighting procedure. The gradient of the time-dependent $CP$-asymmetry, $\Delta Y$, is measured to be \begin{equation} \Delta Y = (-1.3 \pm 6.3\,(\text{stat}) \pm 2.4\, (\text{syst})) \times 10^{-4},\nonumber \end{equation} which is compatible with $CP$ conservation. The decay $D^0 \rightarrow K^- \pi^+ \pi^0$ is used as a control channel to validate the measurement procedure. After correcting for detection asymmetries, the control channel is also found to be compatible with no time-dependent $CP$ asymmetry.