  Reconstructing the density fluctuations in the early Universe that evolved
into the distribution of galaxies we see today is a challenge of modern
cosmology [ref.]. An accurate reconstruction would allow us to test
cosmological models by simulating the evolution starting from the reconstructed
state and comparing it to the observations. Several reconstruction techniques
have been proposed [8 refs.], but they all suffer from lack of uniqueness
because the velocities of galaxies are usually not known. Here we show that
reconstruction can be reduced to a well-determined problem of optimisation, and
present a specific algorithm that provides excellent agreement when tested
against data from N-body simulations. By applying our algorithm to the new
redshift surveys now under way [ref.], we will be able to recover reliably the
properties of the primeval fluctuation field of the local Universe and to
determine accurately the peculiar velocities (deviations from the Hubble
expansion) and the true positions of many more galaxies than is feasible by any
other method.
  A version of the paper with higher-quality figures is available at
http://www.obs-nice.fr/etc7/nature.pdf
