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Our research objective in this paper is to reconstruct an initial lineardensity field, which follows the multivariate Gaussian distribution withvariances given by the linear power spectrum of the current CDM model andevolves through gravitational instability to the present-day density field inthe local Universe. For this purpose, we develop a Hamiltonian Markov ChainMonte Carlo method to obtain the linear density field from a posteriorprobability function that consists of two components: a prior of a Gaussiandensity field with a given linear spectrum, and a likelihood term that is givenby the current density field. The present-day density field can bereconstructed from galaxy groups using the method developed in Wang et al.(2009a). Using a realistic mock SDSS DR7, obtained by populating dark matterhaloes in the Millennium simulation with galaxies, we show that our method caneffectively and accurately recover both the amplitudes and phases of theinitial, linear density field. To examine the accuracy of our method, we use$N$-body simulations to evolve these reconstructed initial conditions to thepresent day. The resimulated density field thus obtained accurately matches theoriginal density field of the Millennium simulation in the density range 0.3 <=rho/rho_mean <= 20 without any significant bias. Especially, the Fourier phasesof the resimulated density fields are tightly correlated with those of theoriginal 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.

Reconstructing the Initial Density Field of the Local Universe: Method and Test with Mock Catalogs