$searchDate = "2013-03-01";

$query->usePublicationQuery()
            ->filterByPublishedAt($searchDate)
            ->endUse();

->filterByPublishedAt(array("min" => $searchDate." 00:00:00", "max" => $searchDate." 23:59:59"))

$c = new Criteria();
$c->add(SomePeer::CREATED_AT, $date.'%', Criteria::LIKE);

$c = new Criteria();
$c->add(SomePeer::CREATED_AT, SomePeer::CREATED_AT.' LIKE "'.$date.'%"',Criteria::CUSTOM);