Reconstructing the Initial Density Field of the Local Universe: Method
  and Test with Mock Catalogs
  Our research objective in this paper is to reconstruct an initial linear
density field, which follows the multivariate Gaussian distribution with
variances given by the linear power spectrum of the current CDM model and
evolves through gravitational instability to the present-day density field in
the local Universe. For this purpose, we develop a Hamiltonian Markov Chain
Monte Carlo method to obtain the linear density field from a posterior
probability function that consists of two components: a prior of a Gaussian
density field with a given linear spectrum, and a likelihood term that is given
by the current density field. The present-day density field can be
reconstructed from galaxy groups using the method developed in Wang et al.
(2009a). Using a realistic mock SDSS DR7, obtained by populating dark matter
haloes in the Millennium simulation with galaxies, we show that our method can
effectively and accurately recover both the amplitudes and phases of the
initial, linear density field. To examine the accuracy of our method, we use
$N$-body simulations to evolve these reconstructed initial conditions to the
present day. The resimulated density field thus obtained accurately matches the
original density field of the Millennium simulation in the density range 0.3 <=
rho/rho_mean <= 20 without any significant bias. Especially, the Fourier phases
of the resimulated density fields are tightly correlated with those of the
original simulation down to a scale corresponding to a wavenumber of ~ 1 h/Mpc,
much smaller than the translinear scale, which corresponds to a wavenumber of ~
0.15 h\Mpc.
