The galaxy power spectrum on the lightcone: deep, wide-angle redshift
  surveys and the turnover scale
  We derive expressions for the survey-window convolved galaxy power spectrum
in real space for a full sky and deep redshift survey, but taking into account
the geometrical lightcone effect. We investigate the impact of using the
standard mean redshift approximation as a function of survey depth, and show
that this assumption can lead to both an overall amplitude suppression and
scale-dependent error when compared to the `true' spectrum. However, we also
show that by using a carefully chosen `effective fixed-time', one can find a
range of scales where the approximation to the full model is highly accurate,
but only on a more restricted set of scales. We validate the theory by
constructing dark matter and galaxy lightcone mock surveys from a large
$N$-body simulation with a high cadence of snapshots. We do this by solving the
light cone equation exactly for every particle, where the particle worldlines
are obtained in a piecewise fashion with cubic interpolation between
neighbouring snapshots. We find excellent agreement between our measurements
and the theory ($\sim \pm 5\%$) over scales $(0.004 \ h \ {\rm Mpc}^{-1} \leq k
\leq 0.54 \ h \ {\rm Mpc}^{-1})$ and for a variety of magnitude limits. We also
test how well the commonly used FKP weights affect the measurements, for
various choices of the fiducial power $P_0$ and apparent magnitude cuts. We
find that including the weighting scheme can boost the signal-to-noise ratio by
factors of a few. Finally, we look to see how accurately we can measure the
turnover scale of the galaxy power spectrum $k_0$. Using the lightcone mocks we
show that one can detect the turnover scale with a probability $P \geq 95\%$ in
an all-sky catalogue limited to an apparent magnitude $m_{\rm lim}\sim 21$. We
also show that the detection significance would remain high for surveys with
$m_{\rm lim}\sim22$ and $20\%$ sky coverage.
