  Detection and parameter inference of gravitational-wave signals \ncor{from
compact mergers} rely on the comparison of the incoming detector strain data
$d(t)$ to waveform templates for the gravitational-wave strain $h(t)$ that
ultimately rely on the resolution of Einstein's equations via numerical
relativity simulations. These, however, commonly output a quantity known as the
Newman-Penrose scalar $\psi_4(t)$ which, under the Bondi gauge, is related to
the gravitational-wave strain by $\psi_4(t)=\mathrm{d}^2h(t) / \mathrm{d}t^2$.
Therefore, obtaining strain templates involves an integration process that
introduces artefacts that need to be treated in a rather manual way. By taking
second-order finite differences on the detector data and inferring the
corresponding background noise distribution, we develop a framework to perform
gravitational-wave data analysis directly using $\psi_4(t)$ templates. We first
demonstrate this formalism, and the impact of integration artefacts in strain
templates, through the recovery of numerically simulated signals from head-on
collisions of Proca stars injected in Advanced LIGO noise. Next, we re-analyse
the event GW190521 under the hypothesis of a Proca-star merger, obtaining
results equivalent to those in Ref.[1], where we used the classical strain
framework. We find, however, that integration errors would strongly impact our
analysis if GW190521 was four times louder. Finally, we show that our framework
fixes significant biases in the interpretation of the high-mass GW trigger
S200114f arising from the usage of strain templates. We remove the need to
obtain strain waveforms from numerical relativity simulations, avoiding the
associated systematic errors.
