  We investigate the effect of long-range scalar interactions in dark matter
(DM) models of cosmic structure formation with a particular focus on the
formation times of haloes. Utilising $N$-body simulations with $512^3$ DM
particles we show that in our models dark matter haloes form substantially
earlier: tracing objects up to redshift $z\sim6$ we find that the formation
time, as characterised by the redshift $z_{1/2}$ at which the halo has
assembled half of its final mass, is gradually shifted from $z_{1/2}\approx
1.83$ in the fiducial LCDM model to $z_{1/2}\approx 2.54$ in the most extreme
self-interaction model. This is accompanied by a shift of the redshift that
marks the transition between merger and steady accretion epochs from
$z_{*}\approx 4.32$ in the \lcdm\ halos to $z_{*}\approx 6.39$ in our strongest
interaction model. In other words, the scalar-interacting model employed in
this work produces more structures at high redshifts, prolonging at the same
time the steady accretion phases. These effects taken together can help the
LCDM model to account for a high redshift reionisation as indicated by the WMAP
data and can alleviate issues related to the survival of the thin-disk
dominated galaxies at low redshifts.
