create DateTime string for the beginning and the end of the day

Use DateTime::setTime

$d = new \DateTime("2013-10-28");
$d->setTime(23, 59, 59);

Have a look at all possible compound formats.

The MySQL format should be the easiest for your use case:

new \DateTime("2013-10-28 00:00:00");
new \DateTime("2013-10-28 23:59:59");

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