  Because of the relatively broad angular resolution of current gamma-ray
instruments in the MeV-GeV energy range, the photons of a given source are
mixed with those coming from nearby sources or diffuse background. This source
confusion seriously hampers the search for pulsation from faint sources.
Statistical tests for pulsation can be made significantly more sensitive when
the probability that a photon comes from the pulsar is used as a weight.
However, the computation of this probability requires knowing the spectral
model of all sources in the region of interest, including the pulsar itself.
This is not possible for very faint pulsars that are not detected as gamma-ray
sources or whose spectrum is not measured precisely enough. Extending the
event-weighted pulsation search to such very faint gamma-ray sources would
allow improving our knowledge of the gamma-ray pulsar population. We present
two methods that overcome this limitation by scanning the spectral parameter
space, while minimizing the number of trials. The first one approximates the
source/background ratio yielding a simple estimate of the weight while the
second one makes use of the full spatial and spectral information of the region
of interest around the pulsar. We test these new methods on a sample of 144
gamma-ray pulsars already detected by the Fermi Large Area Telescope data. Both
methods detect pulsation from all pulsars of the sample, including the ones for
which no significant phase-averaged gamma-ray emission is detected.
