Cosmological constraints from the redshift dependence of the
  Alcock-Paczynski effect: Fourier space analysis
  The tomographic Alcock-Paczynski (AP) method utilizes the redshift evolution
of the AP distortion to place constraints on cosmological parameters. It has
proved to be a robust method that can separate the AP signature from the
redshift space distortion (RSD) effect, and deliver powerful cosmological
constraints using the $\lesssim 40h^{-1}\ \rm Mpc$ clustering region. In
previous works, the tomographic AP method was performed via the anisotropic
2-point correlation function statistic. In this work we consider the
feasibility of conducting the analysis in the Fourier domain and examine the
pros and cons of this approach. We use the integrated galaxy power spectrum
(PS) as a function of direction, $\hat P_{\Delta k}(\mu)$, to quantify the
magnitude of anisotropy in the large-scale structure clustering, and use its
redshift variation to do the AP test. The method is tested on the large, high
resolution Big-MultiDark Planck (BigMD) simulation at redshifts $z=0-1$, using
the underlying true cosmology $\Omega_m=0.3071,\ w=-1$. Testing the redshift
evolution of $\hat P_{\Delta k}(\mu)$ in the true cosmology and cosmologies
deviating from the truth with $\delta \Omega_m=0.1,\ \delta w=0.3$, we find
that the redshift evolution of the AP distortion overwhelms the effects created
by the RSD by a factor of $\sim1.7-3.6$. We test the method in the range of
$k\in(0.2,1.8)\ h\ \rm Mpc^{-1}$, and find that it works well throughout the
entire regime. We tune the halo mass within the range $2\times 10^{13}$ to
$10^{14}\ M_{\odot}$, and find that the change of halo bias results in
$\lesssim 5 \%$ change in $\hat P_{\Delta k}(\mu)$, which is less significant
compared with the cosmological effect. Our work shows that it is feasible to
conduct the tomographic AP analysis in the Fourier space.
