Marginally deformed rolling tachyon around the tachyon vacuum in open string field theory

We investigate the string field theory around the tachyon vacuum. A pure gauge form of the solution is constructed at the tachyon vacuum. For a special choice of the gauge function for the pure gauge form, marginal deformation from the tachyon vacuum is allowed due to the nontrivial roles of Schnabl's analytic vacuum solution. We obtain an exact rolling tachyon solution which describes the late time behaviors of D-brane decay.